Mth 95 Module 1 Spring 2014

Transcription

1 Section 3. - Graphing Equations The Rectangular Coordinate System, which is also called the system. On the grid below locate and label the following: Origin ( ) x-axis ( ) y-axis ( ) Quadrants I, II, III and IV Points plotted in a plane are called because the order in which they are written tells you which number is associated with which axis. Plot each point below and tell the quadrant in which the point lies or the axis on which it lies. In the first point (3,-), 3 is called the and - is called the. Point (3,-) (0,3) (-4,) (-,0) (- ½,-3) Quadrant or axis Solutions of equations in two variables consist of two numbers that form a true statement when substituted into the equation. Determine whether each point is a solution of the equation. y = -x + 7 (, 5) (-, 3) Linear Equations: Standard form (in two variables) The equation for a line is in standard form when it is written as Ax + By = C, where A, B, and C are constants with A and B not both 0. For each of the following examples identify A, B, and C. 5x y = - -x y = 9 y = 5 x + y = 3 Chapter 3

2 A graph is a picture of all the of an equation. To graph a linear equation we only need points. Quick way to Graph a line in Ax + By = C form Find the intercepts of a line algebraically To find the x-intercept of a line, let y = 0 in its equation and solve for x. To find the y-intercept of a line, let x = 0 in its equation and solve for y. Find the x and y intercepts for 4x 3y = -. x-intercept 4x 3(0) = - 4x = - x = -3 y-intercept 4(0) 3y = - -3y = - y = 4 Use these intercepts to sketch a graph of the equation. Be sure to use arrows. Find the intercepts of x + 3y = 6.and then graph it. x-intercept y-intercept Non-Linear Equations: When graphing non-linear equations more points will be needed to create the graph. On the graph of the quadratic equation: y x x 3, each square is units. Locate and label: The vertex: The Axis of Symmetry (AOS): The x-intercept(s): The y-intercept: Note: for an equation with two variables you cannot list all the Chapter 3

3 Fill out the table and sketch the graph of y x x x y The absolute value of a number is the distance of that number from zero. Since distance is always measured in positive units, the absolute value of any number will always be positive. The notation is: y x. See Example 7 on page 4 of your test for the graph of this equation. Graph the equation: y x Graph the equation: y x x y x y Graph the exponential function x y y x Graph the quadratic function y = x x y Chapter 3 3

4 Section 3. - Introduction to Functions Vocabulary of Relations and Functions A is any set of ordered pairs, (x, y). The set of all first components of the ordered pairs is called the or of the relation. The set of all second components is called the or of the relation. A is a relation in which each first component of the ordered pairs, the, corresponds to exactly one second component, the. No two ordered pairs of a function can have the same first component and different second components, that is, each input has a unique output. Relations and functions can be displayed in six ways. For each of the following, state the domain and range and tell whether it is a function. For finite sets give the domain and range in set notation. For infinite sets give the domain and range in interval notation. ) as a set of ordered pairs Domain: 0,,,4,,,,3, 3, ) in a table Input, x Output, y Range: Function: Domain: Range: Function: 3) in a diagram For the relation in the diagram below, list the set of ordered pairs. x y 0 - Domain: Range: Function: Remember a function has only one output for each valid input. 4) function displayed in an equation x 3 y Domain Range Function y x 4 Domain Range Function Chapter 3 4

5 Review function notation: f(x) is read f of x. It represents the function f written in terms of x. y = f(x) so x is the variable, the or and f(x) is the variable, the or Note: f(x) does not mean f times x. 5) function displayed on a graph (See examples 5 and 6 on pages 3 34) When a relation is graphed, we use the vertical line test to determine whether the relation is a function. VLT If no vertical line can be drawn so that it intersects the graph more than once, then the graph represents a function. Do the following graphs display functions? Why or why not? Each square represents unit. 4 x 9y 36 h( x) x 4 Domain Range Function Point A Point B Identify each of the following points on the graph of h(x) above. (-, h(-)) is the point h(-) = (-3, h(-3)) is the point h(-3) = Point C (5, h(5)) is the point h(5) = Point D (0, h(0)) is the point h(0) = For which x-values does h(x) = -3? or 6) Functions expressed in words Write the symbolic description for the area of a triangle whose height is 3 inches more than its base, x. State the domain and range. (Remember the domain of a function is all inputs that produce valid outputs.) Evaluate the function for x = 3 and interpret the results. Chapter 3 5

6 In 008, LBCC is paying $0.48 per mile when a faculty member uses his/her own vehicle for transportation to a sanctioned event. Write a symbolic description for the cost, C(x), of driving x miles. State the domain and range. Find C(30) and interpret what it means. Write the symbolic description for the perimeter (p) of a square with side x. Evaluate for an input of 7cm and describe what it means. Given the function: f ( x ) 3 x 5 find f 3 Given the function: g x x find g and g a ( ) 3 ( ) ( ) The function Pt ( ).08(.039) t represents the population in billions in India where t is the number of years after 005. Find P(0) and P() and describe what they represent. Section 3.3 Graphing Linear Functions Review: Identifying linear functions represented symbolically A function is linear if it can be written in slope-intercept form, f(x) = ax + b (or y = mx + b). Which of the following are linear functions? Why or why not? f(x) = 8 x Yes or No f(x) = 5 Yes or No f(x) = x - Yes or No Chapter 3 6

7 4 3x y Yes or No y x Yes or No x y Yes or No x 6 f x x 3 Yes or No Determining whether the function represented numerically in a table is linear For each unit of increase in the input is there a constant change in the output? x f(x) x 0 3 f(x) Constant change in f(x) = Change in f(x) = Constant change in x = Change in x = Linear? Linear? Determining whether the relation represented in each graph is a linear function. Which of the following graphs represent linear functions (caution!)? Determine whether the following verbal descriptions are of linear functions. If they are, define your variables and write an equation to represent it. Remember that linear functions must increase or decrease at a constant rate. The cost of a number (n) of first class stamps is proportional to the number of stamps you purchase. One first class stamp costs $0.4. The Maytag repairman comes to my house to repair my washing machine. He charges me $40 per hour for the work he does and an additional $35 for the "house call". Chapter 3 7

8 A potent antibacterial medication is so effective it kills about half the bacteria in a wound every day it is being applied. Solving Applications As a gas, like helium, is heated, it will expand. The formula: V(t) = 0.47t + 40 calculates the volume in cubic inches of a sample of helium at the temperature, t, in degrees Centigrade (Celsius). a) EvaluateV(0) and interpret what it means. b) If the temperature increases by 0 degrees, how much does the volume of this sample increase? c) What is the volume of this sample of helium at 00 degrees? d) At what temperature will this sample of helium reach a volume of 50 cubic inches? Slope-intercept form of a line The line with slope m and y-intercept (0, b) is given by Remember the two easiest ways to Graph Lines Slope-intercept: Transform the equation to slope intercept form, plot the y- intercept, use slope to plot other points on the line, and draw the line through the points you ve plotted.. X and Y intercepts: Replace x with 0 to find the y-intercept and y with 0 to find the x-intercepts. Plot these points and draw a line through them. y = 3x + y = -x - f(x) = 3 x Chapter 3 8

9 Find the x and y-intercepts of 4x 3y and use them to sketch the graph. Other Methods to Graph Lines Use a table of values to graph Graph these two equations on the the equation: 3y x 6. To make this same set of axes. What do you easier, solve for y and then choose values notice about the lines? for x that are multiples of 3 to avoid x 3 and y fractions. Which graph is not a function? Section Slope of a Line The slope of a line passing through points (x, y ) and (x, y ) is m = Find the slope of a line that goes through (-3, 6) and (4,-). Slope can be found by counting the rise and run on a graph or using the coordinates of two points in the above formula. If the line rises from If the line falls from Horizontal lines Vertical lines left to right, the slope left to right, the slope always have always have is always is always slope slope. Chapter 3 9

10 Given a point on the line and slope, graph the line. Steps: (, ) m = ½ (-, 0) m = -3 ) Plot the point given ) Use the slope to find and plot a second point 3) Draw the line Given a graph of a linear equation, write the equation in slope-intercept form Equations of horizontal and vertical lines The equation of a horizontal line with y-intercept (0,c) is The equation of a vertical line with x-intercept (c,0) is Write the equation for each line displayed. The scale on both axes is. Identifying the slope and y-intercept in linear equations Identify the slope and y-intercept in each equation. It may be necessary to transform the equation into slope intercept form. f(x) = x 4-3x + y = 5 y = -3 x = 4 Parallel Lines: Two lines are parallel if the have the same but different. Chapter 3 0

11 Perpendicular Lines: Two non-vertical lines with non-zero slopes m and m are perpendicular if the of their slopes is Pairs of perpendicular slopes m - 3/4-0.5 m / On the same set of axes, graph y x 3 y x State whether each pair of equations represent parallel lines, perpendicular lines or neither. (Remember, examine the slopes. Their product must be -) x 3y x 4y 3 4x 6y 3 3x y 5 Slope in applications - Slope can be interpreted as a rate of change of a quantity. In 990 about. million SUV's were sold. In 99 about.4 million were sold. Let x = 0 represent 990. If this trend is linear, write a model to represent it and predict the number of SUV's sold in 000. Chapter 3

12 Section Equations and Linear Models, Parallel and Perpendicular Lines Review Write an equation in slope-intercept form for the line with slope -0.5 and vertical intercept of.75. Does (, 0.75) lie on this line? Find the equations of the vertical line and the horizontal line passing through (,-3). Vertical line Horizontal line Give the parallel and perpendicular slopes to y = -3x + 5, Parallel slope Point-slope form of a linear equation Perpendicular slope A line with slope m passing through (x, y ) is given by or Write an equation in point-slope form for a line with a slope of -3 which goes through the point (,4). Transform the equation into slope-intercept form. Write an equation in point-slope form of a line passing through (-4, -) with a slope of ½. Give the slope and a point on the line: a) y - = 3(x 5) b) y = -3(x 5) + 7 c) y = 0 d) x = -5 e) f(x) = 5x + 3 f) g(x) = 3x Chapter 3

13 Two methods for finding the equation of a line through any two points. (This skill of writing equations for lines through points becomes really important when we solve linear application problems and can describe the information given as an ordered pair.) Method : Using point-slope formula ) Compute slope m ) Substitute into point-slope form 3) Transform the equation to slope-intercept form 4) Check that both points lie on the line Use point-slope form to find the Use the point-slope form to write the slope-intercept form of the equation slope-intercept form of the equation of the line passing through for the line through the points (-, 5) and (4, -). (-3, ) and (,4). Method : Using slope-intercept formula ) Calculate slope m ) Algebraically find the y-coordinate (b) of the y-intercept 3) Using m and b write an equation for the line in slope-intercept form. 4) Check that both points lie on the line Use slope-intercept to find an equation for the line which goes through the points (-3, 5) and (, 4). Write your final equation in slope-intercept form. Finding the equations of a line through a given point that is parallel to a given line Steps: ) Determine the parallel slope ) Use steps through 4 from method or to find the slope-intercept form Find the slope-intercept form of a Find the slope-intercept form of a line parallel to y = x + 5 which line parallel to y x which passes through the point (-, 3). passes through the point (-,4). 3 Chapter 3 3

14 Finding the equations of a line through a given point that is perpendicular to a given line Steps: ) Determine perpendicular slope ) Use steps through 4 from method or to find the slope-intercept form Find the slope-intercept form of a Find the slope-intercept form of a line perpendicular to y = -3x + line perpendicular to y = /3x + which passes through the point (-, 5). which passes through the point (4, ). Application: The projected annual cost of the average private college or university is shown in the table. The cost includes, tuition, fees, room and board. Year Cost $5,000 $40,000 a) Find the slope intercept form of a line that goes through these data points. b) Find another equation for this data by letting x = 0 represent the year 000. c) Compare the equations d) Interpret the slope as a rate of change e) Using you equation from b, estimate the cost of private college in 005. Chapter 3 4