x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =

Size: px
Start display at page:

Download "x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m ="

Transcription

1 Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the slope value is away from zero. Another thing to keep in mind is that the slope is the same value from point to point (or between any two points on the line). This is called the average rate of change because it describes how much the line changes from point to point. There are three different ways to find the slope, labeled m. rise 1) m = this is the graphical way run y y1 ) Given two points ( 1, y 1 ) and (, y ): m = 1 3) Calculate how much y changes from point to point and compare that to how much changes from point to point. The third way requires a little eplanation. Let s say we re given the points (3, 6) and (5, 9) and we want to find what the slope of the line is between them. I m going to make a t chart. means change in and y means change in y, or how much and y change from point to point. y y increases by y increases by y 3 Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m = y y = =. I just prefer and y and will use it almost eclusively this quarter because of its adaptation to polynomials in general. There are a couple of other things to notice about slopes. One is that if a slope is positive, it is going uphill from left to right; if a slope is negative, it is going downhill from left to right. In other words, in a positive slope, as gets bigger, y also gets bigger. In a negative slope, as gets bigger, y gets smaller. Since the slope above is positive, it is going uphill from left to right. Let s graph it! y We can also use the slope to find more points on the same line. Keep in mind that the slope is y = =, so that if we + increase y by 3 and increase by, we get another point!

2 rise Notice how, as we go from point to point on the graph, we can identify the. If we go from run (3, 6) to (5, 9), we get y = +3 and = +, so, again, y = though, see how we get the same thing? This time y = -3 and = -, so This works because of the nature of ratios! 3. If we go from (5, 9) to (3, 6), y 3 = = Practice - Find the slope of the line between the pairs of points. 1) (-, -7) and (1, -1) ) (3, ) and (-3, ) 3) (, 5) and (, 3) 3. y y 3-3 y 5 3 m = y = m = y = m = y = One thing to notice about # is that with a slope of 0, our line is going to be horizontal. See how it s flat and has no incline? In #3, we have an undefined slope, which gives us a vertical line. In this case, our line is so infinitely steep that we don t have a number to represent its incline. We can also eamine the slope of a line based on the given equation of the line. For eample, let f() = 3 +. To find points on this line, we re not going to use the standard -, - 1, 0, 1, like we did for graphing basic equations. If we did that, we d end up having fractions as point values, and that s difficult to graph. It would be easier for us to pick -values that will cancel with that in the denominator. So, let s pick multiples of, like 0,, 8, and so on. f() = y = 3 + Also, eamine the slope from point to point using and y. What is the slope of this line? y y Do you see that number represented in the equation 0 y = 3 +? 8 Another thing to notice is that when = 0, the y-value is. Do you see the number represented in the equation y = 3 +?

3 The form y = m + b (when we solve for y ) is called slope-intercept form because if a line is in this form, the coefficient of is the slope and b is the y-intercept (the y-value when = 0 this is where the line crosses the y-ais). Notice how when we plug in 0 for, the m portion of the equation disappears and we re left with y = b. So, m is our slope and b is our y-intercept. Practice Find the slope and the y-intercept point on the following lines just by eamination: 1) y = 6 1 ) y = + 3 3) y = + 10 ) y = point point point point The neat thing is that once we get one point, we can use the slope to find other points on the rise line, either by using the on the graph or by using and y on the t-chart. This also run means that if we know the y-intercept and the slope, we can find the equation of the line easily by simply plugging into our y = m + b format. Be aware, however, that this method only works if we are given a y-intercept, or a point where the -value = 0. It does NOT work if the point we re given isn t the y-intercept! Practice - Find the equation of the line having the given slope and passing through the given y-intercept. Notice that the given points all have = 0. 1) m = 3, (0, ) ) m =, (0, -3) 9 7 3) m =, (0, 9) ) m = 9, (0, -1) y = y = y = y = Another linear form that s important is what is commonly called standard form. It looks like this: A + By = C. The way we turn standard form into slope-intercept form is simply by solving for y. Practice Write the following in slope-intercept form ( y = m + b form). 1) 3 + y = 8 ) 5 7y = 1 3) + 3y = 7

4 Practice Graphing Lines that are in Slope-Intercept Form The lines below are given in slope-intercept form. Find the slope, y-intercept (b), and then graph the line. 3 1) y = + 3 ) y = 3) y = m = m = m = b = b = b = y y y The advantage to having an equation written in slope-intercept form when graphing is it immediately gives us a point (the y-intercept) and a slope, with which we can count out another point either on our -y grid or by using and y on a t-chart. But what if our line is given to us in standard form (A + By = C)? There are a couple of options. One option is to solve for y and rewrite our equation in slope-intercept form. The other option is to find the - and y-intercepts. In other words, plug in = 0 and find y and then plug in y = 0 and find. Eample: Graph 3 y = 1. Way #1: 3 y = 1 y = 3 3 Way #: 3 y = 1 m = b = 0 y 0 0 y Practice Graphing Lines that are in Standard Form Find the - and y-intercepts and graph the line. Also, use and y to find the slope of each line. 1) 3 + y = 6 ) y = y y

5 3) 5 + y = 10 ) 6 + 3y = 1 y y 5) y = 6 6) 3 + y = 3 y y In general let s solve the standard form of a line (A + By = C) for y so that we can see what it looks like in slope-intercept form. A + By = C What is the slope of the line in this format? Notice the negative sign! This means that we can look at any line in standard form and, just by eamination, figure out the slope. Practice Find the slope of the lines below. 1) 3 + y = 6 ) y = 3) 5 + y = 10 ) 6 + 3y = 1 5) y = 6 6) 3 + y = 3

6 A This method of finding the slope also works in reverse: if we know what the slope of B our line is, we can split the numerator and denominator into its A and B parts and find the front part of the equation of our line. We re looking at that net. Finding Equations of Lines Without a Y-Intercept There are several methods we can use to find the equation of a line. If we are given a slope and a y-intercept (where = 0), then we can simply insert those values into the y = m + b format and be finished. We have used this method previously. But what do we do if the point we re given is not the y-intercept? Well, we re going to use what we learned above, and we re going to avoid fractions! Eamples: Find the beginning of the equation of a line given the following slopes. 1) m = 3 ) m = 3) m = 7 1 ) m = 3 To actually find the entire equation, we need a point to go along with our slope. A slope alone is not enough because there are an infinite number of lines all having the same slope. Eample: The four lines on the grid to the right all have the same slope:. The difference is the value of the actual points the line goes through. We can see that difference most obviously in the values of the y- intercepts. Line #1 (top): y = + y-intercept = (0, ) Line #: y = + 1 y-intercept = (0, 1) Line #3: y = y-intercept = (0, 0) Line # (bottom): y = y-intercept = (0, ) So, if we are given a point in addition to our slope, then we can pinpoint the equation of the line we re looking for. We use the slope to find the beginning of our equation and the point to find the end of it! The eamples below use the same slopes as the first eamples in this section. Eamples: Find the equation of the line through the given point and having the given slope. 1 1) m =, (-, ) ) m =, (5, 6) 3) m =, (-1, -) ) m = 3, (3, -5) 3 7

7 Parallel Lines: The advantage of this method of finding equations of lines is evident where parallel lines are concerned. Parallel lines in a plane are lines that do not cross. They also have the same slope, so the left-hand side of the standard equation will, when simplified, be the same for all parallel lines. Eamples: All five of the following lines are parallel. Find the slopes of each line to prove this. 1 3 y = 3 y = 1 6 y = y = 3 + y = 19 Eample: Find the line parallel to5 y = 10 which passes through the point (-1, ). Perpendicular Lines: Perpendicular lines are lines that cross in a 90-degree angle. In most cases, this means that if one line is going downhill (or has a negative slope), the other will be going uphill (or have a positive slope). The slopes also are reciprocals of each other. If one 3 slope, the slope of a perpendicular line will have a slope of. The only time this doesn t 3 apply is when one of the lines is either horizontal or vertical. In this case, the perpendicular line would be vertical or horizontal, respectively. Once we get the new slope, the rest of the process is the same: Eamples: Find the slopes of the lines perpendicular to the given lines below: 1 3 y = 3 y = 1 6 y = y = 3 + y = 19 Perpendicular slopes: Eample: Find the line perpendicular to5 y = 10 which passes through the point (-1, ).

8 Given two points: Sometimes, instead of being given a slope and a point, we are given two points and are asked to find the equation of the line that passes through those points. While having two points is preferable in graphing, knowing the slope is preferable in finding the equation of the line. So, in this case, we ll have to find the slope first, and then use that slope and one of the two given points to find the rest of the equation. Eample: Find the equation of the line that passes through the points (-, -) and (1, 5). 1) Find the slope: y ) Put the slope into linear form: 3) Use (-, -): OR 3) Use (1, 5): (same answer, yes?) As you can see, the first two parts of this process are the same. Also, it doesn t matter which point you pick to plug in to finish finding your equation. I usually pick the numbers that are smaller and/or positive. If the slope is given to us, we can bypass part 1.

Linear Equations Review

Linear Equations Review Linear Equations Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The y-intercept of the line y = 4x 7 is a. 7 c. 4 b. 4 d. 7 2. What is the y-intercept

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

Section 3.4 The Slope Intercept Form: y = mx + b

Section 3.4 The Slope Intercept Form: y = mx + b Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept Reminding! m = y x = y 2 y 1 x 2 x 1 Slope of a horizontal line is 0 Slope of a vertical line is Undefined Graph a linear

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.

1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved. 1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

More information

Chapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2).

Chapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2). Chapter Section : Equations of Lines Answers to Problems For problems -, put our answers into slope intercept form..* Find the equation of the line with slope, and passing through the point (,0).. Find

More information

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0 LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

More information

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

More information

2.3 Writing Equations of Lines

2.3 Writing Equations of Lines . Writing Equations of Lines In this section ou will learn to use point-slope form to write an equation of a line use slope-intercept form to write an equation of a line graph linear equations using the

More information

Solving Equations Involving Parallel and Perpendicular Lines Examples

Solving Equations Involving Parallel and Perpendicular Lines Examples Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines

More information

2.1 Equations of Lines

2.1 Equations of Lines Section 2.1 Equations of Lines 1 2.1 Equations of Lines The Slope-Intercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is

More information

In this section, we ll review plotting points, slope of a line and different forms of an equation of a line.

In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Math 1313 Section 1.2: Straight Lines In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Graphing Points and Regions Here s the coordinate plane:

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) m = y 2 - y 1 x1 - x 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) m = y 2 - y 1 x1 - x 2 4.4.28 Graphing-Equations of Lines-Slope Interecpt MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) What is the

More information

Additional Examples of using the Elimination Method to Solve Systems of Equations

Additional Examples of using the Elimination Method to Solve Systems of Equations Additional Examples of using the Elimination Method to Solve Systems of Equations. Adjusting Coecients and Avoiding Fractions To use one equation to eliminate a variable, you multiply both sides of that

More information

Alex and Morgan were asked to graph the equation y = 2x + 1

Alex and Morgan were asked to graph the equation y = 2x + 1 Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

More information

The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6

The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6 Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

More information

Lines and Linear Equations. Slopes

Lines and Linear Equations. Slopes Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

More information

Linear Equations and Graphs

Linear Equations and Graphs 2.1-2.4 Linear Equations and Graphs Coordinate Plane Quadrants - The x-axis and y-axis form 4 "areas" known as quadrants. 1. I - The first quadrant has positive x and positive y points. 2. II - The second

More information

Sect The Slope-Intercept Form

Sect The Slope-Intercept Form Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not

More information

Graphing - Parallel and Perpendicular Lines

Graphing - Parallel and Perpendicular Lines . Graphing - Parallel and Perpendicular Lines Objective: Identify the equation of a line given a parallel or perpendicular line. There is an interesting connection between the slope of lines that are parallel

More information

5 $75 6 $90 7 $105. Name Hour. Review Slope & Equations of Lines. STANDARD FORM: Ax + By = C. 1. What is the slope of a vertical line?

5 $75 6 $90 7 $105. Name Hour. Review Slope & Equations of Lines. STANDARD FORM: Ax + By = C. 1. What is the slope of a vertical line? Review Slope & Equations of Lines Name Hour STANDARD FORM: Ax + By = C 1. What is the slope of a vertical line? 2. What is the slope of a horizontal line? 3. Is y = 4 the equation of a horizontal or vertical

More information

GRAPHING LINEAR EQUATIONS IN TWO VARIABLES

GRAPHING LINEAR EQUATIONS IN TWO VARIABLES GRAPHING LINEAR EQUATIONS IN TWO VARIABLES The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms: Slope-Intercept Form: y = mx+ b In an equation

More information

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

More information

A synonym is a word that has the same or almost the same definition of

A synonym is a word that has the same or almost the same definition of Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given

More information

5.1: Rate of Change and Slope

5.1: Rate of Change and Slope 5.1: Rate of Change and Slope Rate of Change shows relationship between changing quantities. On a graph, when we compare rise and run, we are talking about steepness of a line (slope). You can use and

More information

Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown.

Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown. Warm Up Write an equation given the slope and y-intercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and y-intercept From the graph, you can see that the slope is

More information

Graphing Linear Equations

Graphing Linear Equations Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

More information

Rational Functions ( )

Rational Functions ( ) Rational Functions A rational function is a function of the form r P Q where P and Q are polynomials. We assume that P() and Q() have no factors in common, and Q() is not the zero polynomial. The domain

More information

Rational Functions 5.2 & 5.3

Rational Functions 5.2 & 5.3 Math Precalculus Algebra Name Date Rational Function Rational Functions 5. & 5.3 g( ) A function is a rational function if f ( ), where g( ) and h( ) are polynomials. h( ) Vertical asymptotes occur at

More information

Chapter 8. Examining Rational Functions. You see rational functions written, in general, in the form of a fraction: , where f and g are polynomials

Chapter 8. Examining Rational Functions. You see rational functions written, in general, in the form of a fraction: , where f and g are polynomials Chapter 8 Being Respectful of Rational Functions In This Chapter Investigating domains and related vertical asymptotes Looking at limits and horizontal asymptotes Removing discontinuities of rational functions

More information

A Quick Algebra Review

A Quick Algebra Review 1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

More information

Section 1.4 Graphs of Linear Inequalities

Section 1.4 Graphs of Linear Inequalities Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,

More information

Write the Equation of the Line Review

Write the Equation of the Line Review Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections

More information

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

More information

Chapter 4.1 Parallel Lines and Planes

Chapter 4.1 Parallel Lines and Planes Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about

More information

Slope-Intercept Equation. Example

Slope-Intercept Equation. Example 1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine

More information

Creating Equations. Set 3: Writing Linear Equations Instruction. Student Activities Overview and Answer Key

Creating Equations. Set 3: Writing Linear Equations Instruction. Student Activities Overview and Answer Key Creating Equations Instruction Goal: To provide opportunities for students to develop concepts and skills related to writing linear equations in slope-intercept and standard form given two points and a

More information

Name: Class: Date: Does the equation represent a direct variation? If so, find the constant of variation. c. yes; k = 5 3. c.

Name: Class: Date: Does the equation represent a direct variation? If so, find the constant of variation. c. yes; k = 5 3. c. Name: Class: Date: Chapter 5 Test Multiple Choice Identify the choice that best completes the statement or answers the question. What is the slope of the line that passes through the pair of points? 1.

More information

Slope-Intercept Form and Point-Slope Form

Slope-Intercept Form and Point-Slope Form Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.

More information

Graphing Rational Functions

Graphing Rational Functions Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function

More information

Rational functions are defined for all values of x except those for which the denominator hx ( ) is equal to zero. 1 Function 5 Function

Rational functions are defined for all values of x except those for which the denominator hx ( ) is equal to zero. 1 Function 5 Function Section 4.6 Rational Functions and Their Graphs Definition Rational Function A rational function is a function of the form that h 0. f g h where g and h are polynomial functions such Objective : Finding

More information

5. Equations of Lines: slope intercept & point slope

5. Equations of Lines: slope intercept & point slope 5. Equations of Lines: slope intercept & point slope Slope of the line m rise run Slope-Intercept Form m + b m is slope; b is -intercept Point-Slope Form m( + or m( Slope of parallel lines m m (slopes

More information

Coordinate Plane, Slope, and Lines Long-Term Memory Review Review 1

Coordinate Plane, Slope, and Lines Long-Term Memory Review Review 1 Review. What does slope of a line mean?. How do you find the slope of a line? 4. Plot and label the points A (3, ) and B (, ). a. From point B to point A, by how much does the y-value change? b. From point

More information

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

More information

Lesson 9: Graphing Standard Form Equations Lesson 2 of 2. Example 1

Lesson 9: Graphing Standard Form Equations Lesson 2 of 2. Example 1 Lesson 9: Graphing Standard Form Equations Lesson 2 of 2 Method 2: Rewriting the equation in slope intercept form Use the same strategies that were used for solving equations: 1. 2. Your goal is to solve

More information

7.7 Solving Rational Equations

7.7 Solving Rational Equations Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

More information

Objectives: To graph exponential functions and to analyze these graphs. None. The number A can be any real constant. ( A R)

Objectives: To graph exponential functions and to analyze these graphs. None. The number A can be any real constant. ( A R) CHAPTER 2 LESSON 2 Teacher s Guide Graphing the Eponential Function AW 2.6 MP 2.1 (p. 76) Objectives: To graph eponential functions and to analyze these graphs. Definition An eponential function is a function

More information

Example 1. Rise 4. Run 6. 2 3 Our Solution

Example 1. Rise 4. Run 6. 2 3 Our Solution . Graphing - Slope Objective: Find the slope of a line given a graph or two points. As we graph lines, we will want to be able to identify different properties of the lines we graph. One of the most important

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

More information

Writing the Equation of a Line in Slope-Intercept Form

Writing the Equation of a Line in Slope-Intercept Form Writing the Equation of a Line in Slope-Intercept Form Slope-Intercept Form y = mx + b Example 1: Give the equation of the line in slope-intercept form a. With y-intercept (0, 2) and slope -9 b. Passing

More information

COMPARING LINEAR AND NONLINEAR FUNCTIONS

COMPARING LINEAR AND NONLINEAR FUNCTIONS 1 COMPARING LINEAR AND NONLINEAR FUNCTIONS LEARNING MAP INFORMATION STANDARDS 8.F.2 Compare two s, each in a way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example,

More information

Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System. Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

More information

3.4 The Point-Slope Form of a Line

3.4 The Point-Slope Form of a Line Section 3.4 The Point-Slope Form of a Line 293 3.4 The Point-Slope Form of a Line In the last section, we developed the slope-intercept form of a line ( = m + b). The slope-intercept form of a line is

More information

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , ) Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order

More information

Section 4.4 Rational Functions and Their Graphs

Section 4.4 Rational Functions and Their Graphs Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, is a 16 rational function.

More information

Equations of Lines Derivations

Equations of Lines Derivations Equations of Lines Derivations If you know how slope is defined mathematically, then deriving equations of lines is relatively simple. We will start off with the equation for slope, normally designated

More information

Study Guide and Review - Chapter 4

Study Guide and Review - Chapter 4 State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The y-intercept is the y-coordinate of the point where the graph crosses the y-axis. The

More information

Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.

More information

Math 152 Rodriguez Blitzer 2.4 Linear Functions and Slope

Math 152 Rodriguez Blitzer 2.4 Linear Functions and Slope Math 152 Rodriguez Blitzer 2.4 Linear Functions and Slope I. Linear Functions 1. A linear equation is an equation whose graph is a straight line. 2. A linear equation in standard form: Ax +By=C ex: 4x

More information

Answers to Basic Algebra Review

Answers to Basic Algebra Review Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

More information

What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.

What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b. PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

More information

Pre-Calculus III Linear Functions and Quadratic Functions

Pre-Calculus III Linear Functions and Quadratic Functions Linear Functions.. 1 Finding Slope...1 Slope Intercept 1 Point Slope Form.1 Parallel Lines.. Line Parallel to a Given Line.. Perpendicular Lines. Line Perpendicular to a Given Line 3 Quadratic Equations.3

More information

Math Rational Functions

Math Rational Functions Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.

More information

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

Section 3.2. Graphing linear equations

Section 3.2. Graphing linear equations Section 3.2 Graphing linear equations Learning objectives Graph a linear equation by finding and plotting ordered pair solutions Graph a linear equation and use the equation to make predictions Vocabulary:

More information

Graphing Quadratic Functions

Graphing Quadratic Functions Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x

More information

4.4 Concavity and Curve Sketching

4.4 Concavity and Curve Sketching Concavity and Curve Sketching Section Notes Page We can use the second derivative to tell us if a graph is concave up or concave down To see if something is concave down or concave up we need to look at

More information

Introduction to the Practice Exams

Introduction to the Practice Exams Introduction to the Practice Eams The math placement eam determines what math course you will start with at North Hennepin Community College. The placement eam starts with a 1 question elementary algebra

More information

SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. (Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

More information

Learning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations

Learning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations Learning Objectives for Section 1.2 Graphs and Lines After this lecture and the assigned homework, ou should be able to calculate the slope of a line. identif and work with the Cartesian coordinate sstem.

More information

COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

More information

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

More information

Algebra Course KUD. Green Highlight - Incorporate notation in class, with understanding that not tested on

Algebra Course KUD. Green Highlight - Incorporate notation in class, with understanding that not tested on Algebra Course KUD Yellow Highlight Need to address in Seminar Green Highlight - Incorporate notation in class, with understanding that not tested on Blue Highlight Be sure to teach in class Postive and

More information

MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines

MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline 1 Rectangular Coordinate System 2 Graphing Lines 3 The Equation of

More information

Graphing - Slope-Intercept Form

Graphing - Slope-Intercept Form 2.3 Graphing - Slope-Intercept Form Objective: Give the equation of a line with a known slope and y-intercept. When graphing a line we found one method we could use is to make a table of values. However,

More information

Worksheet A5: Slope Intercept Form

Worksheet A5: Slope Intercept Form Name Date Worksheet A5: Slope Intercept Form Find the Slope of each line below 1 3 Y - - - - - - - - - - Graph the lines containing the point below, then find their slopes from counting on the graph!.

More information

How to Find Equations for Exponential Functions

How to Find Equations for Exponential Functions How to Find Equations for Eponential Functions William Cherry Introduction. After linear functions, the second most important class of functions are what are known as the eponential functions. Population

More information

ModuMath Algebra Lessons

ModuMath Algebra Lessons ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations

More information

ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

More information

4.1 & Linear Equations in Slope-Intercept Form

4.1 & Linear Equations in Slope-Intercept Form 4.1 & 4.2 - Linear Equations in Slope-Intercept Form Slope-Intercept Form: y = mx + b Ex 1: Write the equation of a line with a slope of -2 and a y-intercept of 5. Ex 2:Write an equation of the line shown

More information

Section 2.2 Equations of Lines

Section 2.2 Equations of Lines Section 2.2 Equations of Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes

More information

Precalculus A 2016 Graphs of Rational Functions

Precalculus A 2016 Graphs of Rational Functions 3-7 Precalculus A 2016 Graphs of Rational Functions Determine the equations of the vertical and horizontal asymptotes, if any, of each function. Graph each function with the asymptotes labeled. 1. ƒ(x)

More information

Many different kinds of animals can change their form to help them avoid or

Many different kinds of animals can change their form to help them avoid or Slopes, Forms, Graphs, and Intercepts Connecting the Standard Form with the Slope-Intercept Form of Linear Functions Learning Goals In this lesson, you will: Graph linear functions in standard form. Transform

More information

Lesson 6: Linear Functions and their Slope

Lesson 6: Linear Functions and their Slope Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation

More information

Multiplying Polynomials 5

Multiplying Polynomials 5 Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive

More information

Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

More information

2-4 Writing Linear Equations. Write an equation in slope-intercept form for the line described. 2. passes through ( 2, 3) and (0, 1) SOLUTION:

2-4 Writing Linear Equations. Write an equation in slope-intercept form for the line described. 2. passes through ( 2, 3) and (0, 1) SOLUTION: Write an equation in slope-intercept form for the line described 2 passes through ( 2, 3) and (0, 1) Substitute m = 1 and in the point slope form 4 passes through ( 8, 2); Substitute m = and (x, y) = (

More information

Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4)

Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) Chapter 2: Functions and Linear Functions 1. Know the definition of a relation. Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) 2. Know the definition of a function. 3. What

More information

3-4 Equations of Lines

3-4 Equations of Lines 3-4 Equations of Lines Write an equation in slope-intercept form of the line having the given slope and y-intercept. Then graph the 1.m: 4, y-intercept: 3 2. y-intercept: 1 The slope-intercept form of

More information

Plot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line.

Plot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line. Objective # 6 Finding the slope of a line Material: page 117 to 121 Homework: worksheet NOTE: When we say line... we mean straight line! Slope of a line: It is a number that represents the slant of a line

More information

Midterm 1. Solutions

Midterm 1. Solutions Stony Brook University Introduction to Calculus Mathematics Department MAT 13, Fall 01 J. Viro October 17th, 01 Midterm 1. Solutions 1 (6pt). Under each picture state whether it is the graph of a function

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

Situation 2: Undefined Slope vs. Zero Slope

Situation 2: Undefined Slope vs. Zero Slope Situation 2: Undefined Slope vs. Zero Slope Prepared at the University of Georgia EMAT 6500 class Date last revised: July 16 th, 2013 Nicolina Scarpelli Prompt: A teacher in a 9 th grade Coordinate Algebra

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

Equations and Inequalities

Equations and Inequalities Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote

More information