Unit 4 Analyze and Graph Linear Equations, Functions and Relations First Edition Lesson 1 Graphing Linear Equations TOPICS 4.1.1 Rate of Change and Slope 1 Calculate the rate of change or slope of a linear function given information as sets of ordered pairs, a table, or a graph. 2 Apply the slope formula. 4.1.2 Intercepts of Linear Equations 1 Calculate the intercepts of an equation. 2 Use the intercepts to plot a line. 4.1.3 Graphing Equations in Slope Intercept 1 Give the slope intercept form of a linear equation and define its parts. 2 Graph a line using the slope intercept formula and derive the equation of a line from its graph. 4.1.4 Point Slope Form and Standard Form of Linear Equations 1 Give the point slope and standard forms of linear equations and define their parts. 2 Convert point slope and standard form equations into one another. 3 Apply the appropriate linear equation formula to solve problems. Lesson 2 Parallel and Perpendicular Lines TOPICS 4.2.1 Parallel Lines 1 Define parallel lines. 2 Recognize and create parallel lines on graphs and with equations. 4.2.2 Perpendicular Lines 1 Define perpendicular lines. 2 Recognize and create graphs and equations of perpendicular lines. Some rights reserved Monterey Institute for Technology and Education 2011 4.1
4.1.1 Rate of Change and Slope Learning Objective(s) 1 Calculate the rate of change or slope of a linear function given information as sets of ordered pairs, a table, or a graph. 2 Apply the slope formula. Introduction We experience slopes every day. Think about biking down a hill or climbing a set of stairs. Both the hill and the stairs have a slope. That means that as we travel along them, we are moving in two directions at the same time sideways, and either up or down. In conversation, we use words like gentle or steep to describe the slope of the ground or an object. Along a gentle slope, most of the movement is horizontal. Along a steep slope, the vertical movement is greater. Slope Formula The mathematical definition of slope is very similar to our everyday one. In math, slope is the ratio of the vertical and horizontal changes between two points on a surface or a line. The vertical change between two points is called the rise, and the horizontal change is called the run. The slope equals the rise divided by the run:. This simple equation is called the slope formula. The slope formula is often shortened to the phrase rise over run. Although it sounds simple, the slope formula is a powerful tool for calculating and comparing the steepness of landscapes, structures, and lines. 4.2
Self Check A What is the slope formula? A) The vertical change divided by the horizontal change between two points on a line. B) Rise minus run C) The sideways movement over the up or down movement along a line or surface. D) gentle or steep Calculating Slope from a Graph Objective 1 We can find the slope of a line on a graph by counting off the rise and the run between two points. If a line rises 4 units for every 1 unit that it runs, the slope is 4 divided by 1, or 4. A large number like this indicates a steep slope: in this case, the slope goes 4 steps up for every one step sideways. If a line rises 2 units while it runs 8 units, the slope is 2 divided by 8, or!. A small number like this represents a gentle slope. This line goes up only 1 step while it travels 4 steps to the side. The ratio of rise over run describes the slope of all straight lines. This ratio is constant between any two points along a straight line, which means that the slope of a straight line is constant, too, no matter where it is measured along the line. 4.3
Slope Formula and Coordinates It s easy to find the slope of a line on a graph by measuring the rise and the run. We can also find the slope of a straight line if we know the coordinates of any two points on that line. Every point has a set of coordinates: a y-value and an x-value, written as (x, y). The x value tells us where a point is horizontally. The y value tells us where the point is vertically. Consider two random points on a line. We ll call them point 1 and point 2. Point 1 has coordinates (x 1, y 1 ) and point 2 has coordinates (x 2, y 2 ). The rise is the vertical distance between the two points, which is the difference between their y-coordinates. That makes the rise y 2! y 1. The run between these two points is the difference in the x-coordinates, or x 2! x 1. Since slope equals rise over run, the slope of the line is y 2! y 1 over x 2! x 1. We ve now got a new way to write the slope formula and to calculate the value of a slope. Slope is the difference between the y-coordinates divided by the difference between the x-coordinates. Mathematicians commonly use the letter m to represent slope. So this is how you will most often see the slope formula written in algebra:. At heart, though, this is still the simple equation of slope = rise over run. Calculating Slope from Coordinates Objective 2 Let s use the slope formula to calculate the slope of a line, using just the coordinates of two points. The line passes through the points (1, 4) and (-1, 8). Let s call the point (1, 4) point 1, and the point (-1, 8) point 2. That means x 1 = 1, y 1 = 4, x 2 = -1, and y 2 = 8. 4.4
Points Example Coordinates (1, 4) x 1 = 1 y 1 = 4 (-1, 8) x 2 = -1 y 2 = 8 The slope formula is. When we put the coordinates into the formula, we get the equation. This reduces to m = -2. The slope of the line is -2. It is important to realize that it doesn t matter which point is designated as 1 and which is 2. We could have called (-1, 8) point 1, and (1, 4) point 2. In that case, putting the coordinates into the slope formula produces the equation. Once again, that simplifies to m = -2. That s the same slope as before. 4.5
Self Check B What is the slope of the line between the points (-2, 1) and (1, 3)? A) 2/3 B) -2 C) -2/3 D) 3/2 Slope of Horizontal and Vertical Lines So far we ve considered lines that run uphill or downhill. Their slopes may be large or small, but they are always positive or negative numbers. But there are two other kinds of lines, horizontal and vertical. What is the slope of a flat line or level ground? Of a sheer wall or a vertical line? Let s consider a horizontal line on a graph. No matter which two points we choose on the line, they will always have the same y-coordinate. That means the rise, the vertical difference between two points, will always be zero. That makes sense a horizontal line doesn t go up or down. What happens when we put a rise of 0 into the slope formula? It becomes. Zero divided by any number is zero. The slope of a horizontal line is always 0. How about vertical lines? In their case, no matter which two points we choose, they will always have the same x-coordinate, 4.6
That means that the run, the horizontal difference between two points, will always be zero. That makes sense a vertical line doesn t go sideways at all. When we put a run of 0 into the slope formula, the equation becomes. We can t calculate that value because division by zero has no meaning in the set of real numbers. All vertical lines have a slope that is undefined. Summary As you can see, slopes play an important role in our everyday life. You may walk up a slope to get to the bus stop or ski down the slope of a mountain. The slope formula, written, is a useful tool you can use to calculate the vertical and horizontal change of a variety of slopes. 4.7
4.1.1 Self Check Solutions Self Check A What is the slope formula? A) The vertical change divided by the horizontal change between two points on a line. B) Rise minus run C) The sideways movement over the up or down movement along a line or surface. D) gentle or steep A) Correct. The vertical change divided by the horizontal change between two points on a line. B) Incorrect. The correct answer is the vertical change divided by the horizontal change between two points on a line. C) Incorrect. That is the opposite of the slope formula. The correct answer is the vertical change divided by the horizontal change between two points on a line. D) Incorrect. The terms gentle or steep describe a slope verbally, not mathematically. The correct answer is the vertical change divided by the horizontal change between two points on a line. Self Check B What is the slope of the line between the points (-2, 1) and (1, 3)? A) 2/3 B) -2 C) -2/3 D) 3/2 A) Correct. B) Incorrect. The denominator is 1! (-2), not 1! 2. The correct answer is. C) Incorrect. You did not put the coordinates into the slope formula consistently. The answer is. D) Incorrect. You inverted the rise and the run. The correct answer is. 4.8
Learning Objective(s) 1 Calculate the intercepts of a line. 2 Use the intercepts to plot a line. Introduction 4.1.2 Intercepts of Linear Equations The intercepts of a line are the points where the line intercepts, or crosses, the horizontal and vertical axes. The straight line on the graph below intercepts the two coordinate axes. The point where the line crosses the x-axis is called the x-intercept. The y-intercept is the point where the line crosses the y-axis. Notice that the y-intercept occurs where x = 0, and the x-intercept occurs where y = 0. Calculating Intercepts Objective 1 We can use the characteristics of intercepts to quickly calculate them from the equation of a line. Just see how easy it is, as we find the x- and y-intercepts for the line. To find the y-intercept, we substitute 0 for x in the equation, because we know that every point on the y-axis has an x-coordinate of 0. Once we do that, we can solve to find the 4.9
value of y. When we make x = 0, the equation becomes, which works out to y = 2. So when x = 0, y = 2. The coordinates of the y-intercept are (0, 2). Example Problem 3y + 2x = 6 3y + 2(0) = 6 3y = 6 = Answer y = 2 Now we ll follow the same steps to find the x-intercept. We ll let y = 0 in the equation, and solve for x. When y = 0, the equation for the line becomes, and that works out to x = 3. When y = 0, x = 3. The coordinates of the x-intercept are (3, 0). Example Problem 3y + 2x = 6 3(0) + 2x = 6 2x = 6 = Answer x = 3 See, I told you that it would be easy. Self Check A What is the y-intercept of a line with the equation? A) B) (-4, 0) C) (0, -4) D) (5, -4) 4.10
Using Intercepts to Graph Lines Objective 2 Knowing the intercepts of a line is a useful thing. For one thing, it makes it easy to draw the graph of a line we just have to plot the intercepts and then draw a line through them. Let s do it with the equation. We figured out that the intercepts of the line this equation represents are (0, 2) and (3, 0). That s all we need to know: And there we have the line. Intercepts and Problem-Solving Intercepts are also valuable tools for predicting or tracking a process. At each intercept, one of the two quantities being plotted reaches zero. That means that the intercepts of a line can be used to mark the beginning and the end of a process. Imagine a student named Morgan who is buying a laptop for $1,080 to use for school. Morgan is going to use the computer store s finance plan to make this purchase she ll pay $45 per month for 24 months. She wants to know how much she will still owe after each month of the plan. She can keep track of her debt by making a graph. The x-axis will be the number of months and the y-axis will represent the amount of money she still owes on the finance plan. Morgan knows two points in her pay-off schedule. The day she buys the computer, she ll be at 0 months passed and $1,080 owed. The day she pays it off completely, she ll be at 24 months passed and $0 owed. With these two points, she can draw a line, running from the y-intercept at (0, 1080) to the x-intercept at (24, 0). 4.11
Morgan can now use this graph to figure out how much money she still owes after any number of months. Let s look at another situation involving intercepts, this time when we know only one intercept and want to find the other. Joe is a lifeguard at the local swimming pool. It s the end of the summer, and the pool is being drained. Joe has to wait by the pool until it s completely empty, so no one falls in and drowns. How can poor Joe figure out how long that s going to take? If Joe has taken an algebra course, he s got it made. The pool contains 10,200 gallons of water. It drains at a rate of 680 gallons per hour. Joe can use that information to make a table of how much water will be left in the pool hour by hour. x, Time (hours) y, Volume of Water (gallons) 0 10,200 1 9,520 2 8,840 3 8,160 4 7,480 Once he s calculated a few data points, Joe can use a graph and intercepts as a shortcut to find out how long it will be until the pool is dry. Joe s starting point is the y- 4.12
intercept, where the pool is full at 10,200 gallons and the elapsed time is 0. Next, he plots the volume of the pool at 1, 2, 3, and finally 4 hours. Now all Joe needs to do is connect the points with a line, and then extend the line until it meets the x-axis. The line intercepts the x-axis when x = 15. So now Joe knows the pool will take 15 hours to drain completely. It s going to be a long day. Summary We ve now seen the usefulness of the intercepts of a line. When we know where a line crosses the x- and y-axes, we can easily produce the graph or the equation for that line. When we know one of the intercepts and the slope of a line, we can find the beginning or predict the end of a process. 4.13
4.1.2 Self Check Solutions Self Check A What is the y-intercept of a line with the equation? A) B) (-4, 0) C) (0, -4) D) (5, -4) A) Incorrect. is the x-intercept. At the y-intercept, x = 0. When 0 is substituted for x in the equation, y = -4. The correct answer is (0, -4). B) (-4, 0) Incorrect. This answer switches the values of x and y. Coordinates are given in the order (x, y). At the y-intercept, x = 0. When 0 is substituted for x in the equation, y = -4. The correct answer is (0, -4). C) (0, -4) Correct. At the y-intercept, x = 0. When 0 is substituted for x in the equation, y = -4. D) (5, -4) Incorrect. This is the co-efficient of x and the constant, not the y-intercept. At the y- intercept, x = 0. When 0 is substituted for x in the equation, y = -4. The correct answer is (0, -4). 4.14
4.1.3 Graphing Equations in Slope Intercept Form Learning Objective(s) 1 Give the slope intercept form of a linear equation and define its parts. 2 Graph a line using the slope intercept formula and derive the equation of a line from its graph. Introduction Straight lines are produced by linear functions. That means that a straight line can be described by an equation that takes the form of the linear equation formula,. In the formula, y is a dependent variable, x is an independent variable, m is a constant rate of change, and b is an adjustment that moves the function away from the origin. In a more general straight line equation, x and y are coordinates, m is the slope, and b is the y-intercept. Because this equation describes a line in terms of its slope and its y-intercept, this equation is called the slope-intercept form. Slope Intercept Formula Objective 1 Given a linear equation in the form, if we change m we change the slope, or steepness, of the line. As m, the slope, gets larger, the line gets steeper. When m gets smaller, the slope flattens. If we change b, we change the y-intercept. A positive y- intercept means the line crosses the y-axis above the origin, while a negative y-intercept means that the line crosses below the origin. Simply by changing the values of m and b, we can define any straight line. That s how powerful and versatile the slope intercept formula is. Self Check A How is the x-intercept represented in the slope intercept form of a linear equation? A) It is represented by x. B) It is represented by m. C) It is represented by b. D) It is not represented. From Graph to Equation Objective 2 Now that we understand the slope intercept form, we can look at the graph of a line and write its equation just from identifying the slope and the y-intercept. Let s try it with this line: 4.15
The slope intercept form is. For this line, the slope is, and the y-intercept is 4. If we put those values into the formula, we get the equation. That s the slope intercept equation of our line. Self Check B What is the equation of the line in the graph below? 4.16
A) B) C) D) From Equation to Graph We ve seen that it s not difficult to convert the graph of a line to an equation. We can also work the other way and produce a graph from a slope intercept equation. Consider the equation. This equation tells us that the y-intercept is at -1. We ll start by plotting that point, (0, -1), on a graph. The equation also tells us that the slope of this line is -3. So we ll count up 3 units and over -1 unit and plot a second point. (We could also have gone down 3 and over +1.) Then we draw a line through both points, and there it is, the graph of. 4.17
Self Check C Which graph shows the line? A) B) C) D) Summary The slope intercept form of a linear equation is written as, where m is the slope and b is the value of y at the y-intercept. Because we only need to know the slope and the y-intercept to write this formula, it is fairly easy to derive the equation of a line from a graph and to draw the graph of a line from an equation. 4.18
4.1.3 Self Check Solutions Self Check A How is the x-intercept represented in the slope intercept form of a linear equation? A) It is represented by x. B) It is represented by m. C) It is represented by b. D) It is not represented. A) It is represented by x. Incorrect. x is an x-coordinate, but not the x-intercept. The slope intercept form of a linear equation is based on the slope and the y-coordinate at the y-intercept. The correct answer is that it is not represented. B) It is represented by m. Incorrect. m is the slope of the line. The slope intercept form of a linear equation is based on the slope and the y-coordinate at the y-intercept. The correct answer is that it is not represented. C) It is represented by b. Incorrect. b is the y-coordinate at the y-intercept. The slope intercept form of a linear equation is based on the slope and the y-coordinate at the y-intercept. The correct answer is that it is not represented. D) It is not represented. Correct. The slope intercept form of a linear equation is based on the slope and the y- coordinate at the y-intercept. Self Check B What is the equation of the line in the graph below? 4.19
A) B) C) D) A) Incorrect. You have inverted the slope. The correct slope of this line is and the y- intercept is -3. The correct answer is. B) Correct. The slope of this line is and the y-intercept is -3. C) Incorrect. 4 is the x-intercept, not the slope. The slope of this line is and the y- intercept is -3. The correct answer is. D) Incorrect. The slope is positive and the y-intercept is negative, not the other way around. The correct answer is. Self Check C 4.20
Which graph shows the line? A) B) C) D) A) Graph A Correct. This line has a positive y-intercept and a steep positive slope, as the equation requires. B) Graph B Incorrect. This line has a gentle slope, while the equation specifies a steep slope. The correct answer is Graph A. C) Graph C Incorrect. This line has a negative y-intercept and a negative slope, while the equation specifies a positive y-intercept and a steep positive slope. The correct answer is Graph A. D) Graph D Incorrect. This line has a negative y-intercept and a gentle slope, while the equation specifies a positive y-intercept and a steep positive slope. The correct answer is Graph A. 4.21
4.1.4 Point Slope Form and Standard Form of Linear Equations Learning Objective(s) 1 Give the point slope and standard forms of linear equations and define their parts. 2 Convert point slope and standard form equations into one another. 3 Apply the appropriate linear equation formula to solve problems. Introduction Linear equations can take several forms, such as the point-slope formula, the slopeintercept formula, and the standard form of a linear equation. These forms allow mathematicians to describe the exact same line in different ways. This can be confusing, but it s actually quite useful. Consider how many different ways you could write a request for milk on a shopping list. You could ask for white milk, cow s milk, a quart of milk, or skim milk, and each of these phrases would describe the exact same product. The description you use will depend on the characteristics that matter most to you. Equations for describing lines can be chosen the same way they can be written and manipulated based upon on which characteristics of the line are of interest. Even better, when a different characteristic becomes important, linear equations can be converted from one form to another. Point Slope Form Objective 1 One type of linear equation is the point slope form, which gives the slope of a line and the coordinates of a point on it. The point slope form of a linear equation is written as. In this equation, m is the slope and (x 1, y 1 ) are the coordinates of a point. Let s look at where this point-slope formula comes from. Here s the graph of a generic line with two points plotted on it. 4.22
The slope of the line is rise over run. That s the vertical change between the two points (the difference in the y-coordinates) divided by the horizontal change over the same segment (the difference in the x-coordinates). This can be written as. This equation is the slope formula. Now let s say that one of these points is a generic point (x, y), which just means it could be anywhere on the line, and the other point is a specific point,. If we plug these coordinates into the formula, we get. Now we can rearrange the equation a little bit by multiplying both sides of the formula by.. This simplifies to is the point-slope formula. We ve converted the slope formula into the point slope formula. We didn t do that just for fun, but because the point slope formula is sometimes more useful than the slope formula, for example when we need to find the equation of a line when given a point and the slope. Let s do an example. Consider a line that passes through the point (1, 3) and has a slope of. 4.23
Putting these values into the point-slope formula, we get. That s the equation of the line. Self Check A Which of the following points lies on the line (y + 8) = 7(x! 5)? A) (5, -8) B) (5, 8) C) (8, 5) D) (8, -5) Standard Form Objective 2 Remember, the point-slope formula is only one type of linear equation. It is effective in describing some of the characteristics of a straight line. However, point slope equations can be awkward to use in some algebraic operations. In such cases, it may be helpful to convert the equation into a different form, the standard form. The standard form of an equation is Ax + By = C. In this kind of equation, x and y are variables and A, B, and C are integers. 4.24
We can convert a point slope equation into standard form by moving the variables to the left side of the equation. Let s go back to that point-slope equation of. We can rearrange the terms as follows: Example Problem (y! 3) = 4(y! 3) = 4y! 12 = -1x + 1 x + 4y! 12 = -x + 1 + x X + 4y! 12 = 1 x + 4y! 12 + 12 = 1 + 12 Standard Form x + 4y = 13 When we shift the variable terms to the left side of the equation and everything else to the right side, we get. This equation is now in standard form. Matching the Formula to the Situation Objective 3 We now know how to convert equations from point slope to standard form, and how to go back and forth between a graph and a linear equation. But with so many choices, how do we decide which form to use in a real-life situation? The answer is to identify what you know and what you want to find out, and see which form uses those terms. Let s look at a situation where one form of an equation is more useful than the others. Andre wants to buy an MP3 player. He got $50 for his birthday, but the player he wants costs $230, so he s going to have to save up the rest. His plan is to save $30 a month until he has the money he needs. We ll help him out by writing an equation to analyze this situation. This will help us to figure out when he will have saved up enough to buy the MP3 player. When we write the equation, we ll let x be the time in months, and y be the amount of money saved. After 1 month, Andre has $80. That means when x = 1, y = 80. So we know the line passes through the point (1, 80). Also, we know that Andre hopes to save $30 per month. This means the rate of change, or slope, is 30. We have a point and we have a slope that s all we need to write a point slope formula, so that s the form of linear equation we ll use. Remember, the point slope form is 4.25
becomes.. When we substitute in Andre s point and slope, the equation Okay, now what? Well, we have a formula that describes Andre s savings plan. We can use that to figure out how long it will take him to save all the money he needs to buy the MP3 player. Remember, the y in this equation represents the amount Andre has saved, and the x represents the number of months he has been saving. We want to find what the value of x is when y equals 230. So we just need to set y equal to 230 in our equation, and solve for x. Example Problem y! 80 = 30(x! 1) 230! 80 = 30(x! 1) 150 = 30x! 30 180 = 30x Answer 6 = x The result is x = 6. It will take Andre 6 months to save the $230 he needs to buy the MP3 player. Because the problem told us that we knew a point and a slope, we were able to choose the right form for the job of writing an equation. Once we wrote the equation, we were able to solve it for the variable we wanted to find. Summary We ve learned that linear equations can be written in different forms, depending upon what we either know or want to know about a line. The point slope form,, is useful in situations involving slope and the location of one or more points. The standard form, Ax + By = C, is usually easier to use when we need to make algebraic calculations. When needs or knowledge change, we can convert an equation from one form into another. 4.26
4.1.4 Self Check Solutions Self Check A Which of the following points lies on the line (y + 8) = 7(x! 5)? A) (5, -8) B) (5, 8) C) (8, 5) D) (8, -5) A) (5, -8) Correct. The point slope formula is y 1 is -8. That means that the correct answer is (5, -8).. In the given equation, x 1 is 5 and B) (5, 8) Incorrect. You ve missed a sign. The point slope formula is given equation, x 1 is 5 and y 1 is -8. That means that the correct answer is (5, -8). C) (8, 5) Incorrect. You ve reversed x and y and missed a sign. The point slope formula is. In the given equation, x 1 is 5 and y 1 is -8. That means that the correct answer is (5, -8).. In the D) (-8, 5) Incorrect. You ve reversed x and y. The point slope formula is given equation, x 1 is 5 and y 1 is -8. That means that the correct answer is (5, -8).. In the 4.27
4.2.1 Parallel Lines Learning Objective(s) 1 Define parallel lines. 2 Recognize and create parallel lines on graphs and with equations. Introduction Objective 1 Parallel lines are two or more lines that never intersect. Examples of parallel lines are all around us, in the two sides of this page and in the shelves of a bookcase. When you see lines or structures that seem to run in the same direction, never cross one another, and are always the same distance apart, there s a good chance that they are parallel. In algebra, we use something more precise than appearance to recognize and create parallel lines. We use equations. Recognizing Parallel Lines Objective 2 Let s look at two parallel lines on a graph. Line A has an equation of. Line B has an equation of. Can you identify what these two lines have in common? 4.28
Try comparing the coefficients of x in each equation. They are both 3. Because the equations are written in slope-intercept form of, the coefficient of x is the slope of the lines. Since both Line A and B have a slope of 3, they have the same slope. We ve discovered a relationship that is true for all parallel lines lines are parallel if they have the same slope. Not convinced? Look at the lines on this graph: The blue lines are parallel they run in the same direction, keep the same distance apart, and never touch and they have the same slope. All three red lines share a slope, and they too are parallel to one another. The red and blue lines are clearly not parallel, and they have different slopes. Creating Parallel Lines To create a parallel line from an equation, we start by identifying the slope of the original line. Then we write a second equation, repeating the slope but changing the y-intercept. Once again, because there are an infinite number of possible y-intercepts, there are an infinite number of equations we could create. 4.29
Self Check A A line has the equation. Which of the following lines is parallel to it? A) B) C) D) Summary Parallel lines are lines that have the same slope. These lines never intersect and always maintain the same distance apart. To create a parallel line from an equation, we simply change the y-intercept. 4.30
4.2.1 Self Check Solutions Self Check A A line has the equation. Which of the following lines is parallel to it? A) B) C) D) A) Incorrect. This line has a different slope than the given line, while parallel lines have the same slope. The correct answer is. B) Correct. Parallel lines have the same slope. C) Incorrect. This is the same line as the one given in the original equation. Parallel lines have the same slope, but are not the same line. The correct answer is. D) Incorrect. This line has the opposite slope of the given line, and parallel lines have the exact same slope. The correct answer is. 4.31
4.2.2 Perpendicular Lines Learning Objective(s) 1 Define perpendicular lines. 2 Recognize and create graphs and equations of perpendicular lines. Introduction Objective 1 Perpendicular lines are two or more lines that intersect at a 90-degree angle, like the two lines drawn on this graph, and the x and y axes that orient them. Perpendicular lines are everywhere, not just on graph paper but also in the world around us, from the crossing pattern of roads at an intersection to the colored lines of a plaid shirt. In our daily lives, we may be happy to call two lines perpendicular if they merely seem to be at right angles to one another. In algebra, we use equations to make sure of it. Recognizing Perpendicular Lines Objective 2 Let s compare the equations of lines that are perpendicular and those that are not to see if there is a pattern that will allow us to recognize perpendicular lines. We ll start with two lines that we know are not perpendicular. 4.32
Line A has an equation of, and Line B has an equation of. These two lines are clearly not at 90-degrees to one another. Do you see any connection between their equations? If you couldn t find anything you re right. There isn t any link between the slopes or y- intercepts of these lines. Now let s look at two lines that are perpendicular: 4.33
On this graph, Line A has the equation of and Line B has the equation of. Do you see any connection between these equations? If you said no again this time, you missed something. It s a little tricky, but there is a relationship between the slopes of these two lines. The slope of Line A is, and the slope of Line B is. These numbers are opposite reciprocals. An opposite reciprocal is the fraction form of a number flipped upside down (that s the reciprocal part), and with its sign changed (that s the opposite part). Write those slopes side by side and it becomes more obvious: and are opposite reciprocals. This is the defining feature of the equations of perpendicular lines the slopes of perpendicular lines are opposite reciprocals. 4.34
Self Check A If a line has a slope of, what is the slope of a line perpendicular to it? A) B) C) D) Creating Perpendicular Lines To create a perpendicular line from an equation, we once again start by identifying the slope of the original line. Then we write a second equation, changing the slope to the opposite reciprocal of the slope in the original equation. Self Check B Which of the following lines are perpendicular to the line? A) and B) and C) D) all of the lines are perpendicular 4.35
Summary Perpendicular lines are lines that cross one another at a 90 angle. They have slopes that are opposite reciprocals of one another. Unlike parallel lines that never touch, perpendicular lines must intersect. 4.2.2 Self Check Solutions Self Check A If a line has a slope of, what is the slope of a line perpendicular to it? A) B) C) D) A) Incorrect. This is the same slope as the original line. Perpendicular lines have opposite reciprocal slopes. The correct answer is. B) Incorrect. Perpendicular lines have opposite reciprocal slopes. This slope is the opposite but not the opposite reciprocal. The correct answer is. C) Incorrect. Perpendicular lines have opposite reciprocal slopes. This slope is the reciprocal but not the opposite reciprocal. The correct answer is. 4.36
D) Correct. Perpendicular lines have opposite reciprocal slopes. is the opposite reciprocal of. Self Check B Which of the following lines are perpendicular to the line? A) and B) and C) D) all of the lines are perpendicular A) and 4.37
Correct. These lines both have a slope of, which I the opposite reciprocal of the slope of 7 in the original equation. Both these lines are perpendicular to the original line. B) and Incorrect. Perpendicular lines have opposite reciprocal slopes. Both these lines have reciprocal slopes to the slope of 7 in the original equation, but only the first line is the opposite reciprocal. The correct answer is and. C) Incorrect. Perpendicular lines have opposite reciprocal slopes. This slope is the opposite of the original slope of 7, but not the opposite reciprocal. The correct answer is and. D) all of the lines are perpendicular Incorrect. Perpendicular lines have opposite reciprocal slopes. The original line has a slope of 7, so all perpendicular lines must have a slope of. The correct answer is and. 4.38
Unit Recap 4.1.1 Rate of Change and Slope As you can see, slopes play an important role in our everyday life. You may walk up a slope to get to the bus stop or ski down the slope of a mountain. The slope formula, written, is a useful tool you can use to calculate the vertical and horizontal change of a variety of slopes. 4.1.2 Intercepts of Linear Equations We ve now seen the usefulness of the intercepts of a line. When we know where a line crosses the x- and y-axes, we can easily produce the graph or the equation for that line. When we know one of the intercepts and the slope of a line, we can find the beginning or predict the end of a process. 4.1.3 Graphing Equations in Slope Intercept Form The slope intercept form of a linear equation is written as, where m is the slope and b is the value of y at the y-intercept. Because we only need to know the slope and the y-intercept to write this formula, it is fairly easy to derive the equation of a line from a graph and to draw the graph of a line from an equation. 4.1.4 Point Slope Form and Standard Form of Linear Equations We ve learned that linear equations can be written in different forms, depending upon what we either know or want to know about a line. The point slope form,, is useful in situations involving slope and the location of one or more points. The standard form, Ax + By = C, is usually easier to use when we need to make algebraic calculations. When needs or knowledge change, we can convert an equation from one form into another. 4.2.1 Parallel Lines Parallel lines are lines that have the same slope. These lines never intersect and always maintain the same distance apart. To create a parallel line from an equation, we simply change the y-intercept. 4.2.2 Perpendicular Lines Perpendicular lines are lines that cross one another at a 90 angle. They have slopes that are opposite reciprocals of one another. Unlike parallel lines that never touch, perpendicular lines must intersect. 4.39
Glossary coordinates intercept linear equation parallel lines perpendicular lines point-slope formula rise run A pair of numbers that identifies a point on the coordinate plane the first number is the x-value and the second is the y-value. A point where a line meets or crosses a coordinate axis. An equation that describes a straight line. Lines that have the same slope and different y-intercepts. Lines that have opposite reciprocal slopes. A form of linear equation, written as is the slope and (x 1, y 1 ) are the co-ordinates of a point. Lines that have opposite reciprocal slopes. Horizontal change between two points., where m slope formula The equation for the slope of a line, written as, where m is the slope and (x 1, y 1 ) and (x 2, y 2 ) are the coordinates of two points on the line. slope slope-intercept form slope-intercept formula standard form of a linear equation x-intercept, x- intercepts y-intercept, y- intercepts The ratio of the vertical and horizontal changes between two points on a surface or a line. A linear equation, written in the form y = mx + b, where m is the slope and b is the y-intercept. A linear equation, written as y = mx + b, where m is the slope and b is the y-intercept A linear equation, written in the form Ax + By = C, where x and y are variables and A, B, and C are integers. The point where a line meets or crosses the x-axis. The point where a line meets or crosses the y-axis. 4.40