Algebra I Notes Linear Functions & Inequalities Part I Unit 5 UNIT 5 LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES

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1 UNIT LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES PREREQUISITE SKILLS: students must know how to graph points on the coordinate plane students must understand ratios, rates and unit rate VOCABULARY: relation: a set of ordered pairs function: a special relation that has a rule that establishes a mathematical relationship between two quantities, called the input and the output. For each input, there is exactl one output domain: the collection of all input values range: the collection of all output values independent variable: the variable in a function with a value that is subject to choice dependent variable: the variable in a relation with a value that depends on the value of the independent variable (input) function notation: a wa to name a function that is defined b an equation. In function notation, the in the equation is replaced with f(x) rate of change: a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable rise: the difference in the -values of two points on a line run: the difference in the x-values of two points on a line slope: the ratio of rise to run for an two points on a line x-intercept: the x-coordinate of the point where the graph intersects the x-axis. -intercept: the -coordinate of the point where the graph intersects the -axis. SKILLS: find rates of change and slopes relate a constant rate of change to the slope of a line Graph linear equations that are in standard or slope-intercept form Graph linear inequalities in two variables Write linear equations in all three forms STANDARDS: F.IF.A.b Recognize situations in which one quantit changes at a constant rate per unit interval relative to another. *(Modeling Standard) A.REI.D.0 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Linear Functions & Inequalities Part I page of 4 /0/

2 F.IF.C.7a F.LE.A. F.IF.B.4- F.IF.C.7a N.Q.A. F.IF.B.6- Graph linear and quadratic functions and show intercepts, maxima, and minima. *(Modeling Standard) Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). *(Modeling Standard) For a linear, exponential, or quadratic function that models a relationship between two quantities, interpret ke features of graphs and tables in terms of the quantities, and sketch graphs showing ke features given a verbal description of the relationship. Ke features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; smmetries; and end behavior. *(Modeling Standard) Graph linear and quadratic functions and show intercepts, maxima, and minima. *(Modeling Standard) Use units as a wa to understand problems and to guide the solution of multi-step problems; choose and interpret units consistentl in formulas; choose and interpret the scale and the origin in graphs and data displas. Calculate and interpret the average rate of change of a linear, exponential, or quadratic function (presented smbolicall or as a table) over a specified interval. Estimate the rate of change from a graph of a function over a specified interval. *(Modeling Standard) A.CED.A.- Create linear, exponential, and quadratic equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales. Limit exponentials to have integer inputs onl. *(Modeling Standard) A-CED.A.- Represent constraints b linear equations or inequalities, and b sstems of linear equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. *(Modeling Standard) A.REI.D.- Graph the solutions to a linear inequalit in two variables as a half-plane (excluding the boundar in the case of a strict inequalit), and graph the solution set to a sstem of linear inequalities in two variables as the intersection of the corresponding half-planes. Linear Functions & Inequalities Part I page of 4 /0/

3 LEARNING TARGETS:. To identif a linear function from a table, graph, or equation.. To use intercepts to graph linear functions in standard form.. To relate constant rate of change and slope in linear relationships..4 To graph linear equations using slope intercept form.. To graph linear inequalities in two variables..6 To create a linear equation in slope-intercept form..7 To create a linear equation in point-slope form..8 To create a linear equation in standard form. BIG IDEA: The concept of slope is important because it is used to measure the rate at which changes are taking place. In real-life problems, we often need to explore and understand how things change and about how one item changes in response to a change in another item. Describe the similarities and differences between equations and inequalities including solutions and graphs. Linear functions can be created from various forms of information. The solutions to real-word problems can be found b modeling them with equations and graphs. Notes, Examples and Exam Questions Units.,. To identif a linear function from a table, graph, or equation and To relate constant rate of change and slope in linear relationships. The slope of a line is the constant rate of change occurring as ou move along the line from left to right (the steepness of the line). There are four tpes of slope: positive negative zero undefined as x increases, increases as x increases, decreases as x increases, is constant x is constant Slope of a Line: the number of units the line rises or falls for each unit of horizontal change. horizontal change rise change in m vertical change run change in x x x x x where, and, x x are two points on the line. Note: Slope is a rate of change. It determines the steepness of a line. Linear Functions & Inequalities Part I page of 4 /0/

4 Finding slope given a graph: Ex : Find the slope of the given line. Step One: find two points that fall on the line. Step Two: find the ratio of vertical change to horizontal change (from left to right) between the two points. m = rise run = verticalchange horizontalchange m = rise run = Find the slope of a line using Slope Formula. If ou don t have a graph of the line, ou can find the slope of the line using the slope formula when given two points; ( x, ) and ( x, ): change in m changeinx x x Ex : Find the slope of a line that goes through the points (, 4) and (, 7). Substitute in the given values in the ordered pairs into the slope formula and simplif. m change in 7 (4) change in x () Note: Do not express slope as a mixed number. Leave it in simplified fraction form. *How is the slope like unit rate? An slope is a ratio comparing the change in to the change in x. A rate is a unit rate if it has a denominator of. In the examples above, the slope is ; which tells us that as increases b three units, x increases b two units. But, another wa to look at is that as increases b units, x increases b one unit. Linear Functions & Inequalities Part I page 4 of 4 /0/

5 Let s look at the line in the first example. As x increases b unit (from to 4), increases b. units (from 4 to.) Let s look at a table of values for some points that fall on the graph of that linear function. Notice that each time x increases b unit, increases b. units. x The slope is the unit rate of the function. It shows the rate of change verticall, as the graph moves one unit to the right. Ex : Use the table of values to find the rate of change and then explain its meaning. number of video games x total cost ($) Step One: select an two points to use in the slope formula. (, 78) and (4, 6) 6 (78) 78 Step Two: calculate the slope m 9 4 () Step Three: explain what it means in the context of the problem. It means that $9 is the cost per game or the unit rate. Linear Functions & Inequalities Part I page of 4 /0/

6 Ex 4: Determine if the function is linear. Explain our answer. Step One: Pick two points and find the slope between them. x Step Two: (, 0) and (, ) (0) m ( ) Pick two other points and find the slope between them. (, ) and (, 6) 6 () 4 m ( ) Step Three: Determine if it s linear. It is not linear because the slope is not constant. Wh does the slope have to be constant in a linear equation? Because, if the slope is not constant than it is not a linear function. Let s look at that on a graph. Notice that if we start at the point ( 7, ) and follow a slope of, it takes us to the point ( 4, 4). If we follow the same slope again, we get to the point (, ) and then to the point (, ). All of these points fall on a straight line. But, if we change the slope (pattern) to 4, we do not get to a point that falls on the same line as all of the other points. Comparing slopes: Ex : Look at the graph below. Using the line with a slope of as the original line, describe how the steepness of the line changes as the slope changes. 4 m m m = 0 m= *The line is much steeper than the original line. It is rising 4 times quicker while running the same amount. *The line is less steep than the original line. It is rising half as quickl, and running twice as far. *The line is flat. It is not rising at all as it runs to the right. *The line has the same steepness as the original line, except it is going downward. Linear Functions & Inequalities Part I page 6 of 4 /0/

7 Relating Slope to Similar Triangles: Similar triangles are proportional. *All corresponding ratios are proportional. 4 is to 8 as is to Let s put the similar triangles on a coordinate plane. Notice that both triangles hpotenuses fall on the same straight line. The legs of each triangle are like the rise and the run of the slope of the line. The purple triangle s rise is 4 and its run is 8. The red triangle s rise is and its run is 4. Which means that both slopes reduce to an equivalent slope of. *Using similar triangles we can prove that no matter what two points we choose on a line, we will alwas find the same slope between those points. Ex 6: Find the value of so that the line passing through the points (,) and (4, ) has a slope of. Step One: write the slope formula; m x x Step Two: plug in the given values into the formula; () 4 ( ) Step Three: solve for ; = = 6 6 Linear Functions & Inequalities Part I page 7 of 4 /0/

8 Ex 7: Find the slope of the line that passes through the points, and, 4. We will use the slope formula m x x, where x, =, and x,, 4. Substituting into the formula: m 7 On Your Own: If ou change the order of the points, will ou get the same slope? Tr it! Review: Find the quotients. 0? 4? 0 Note: because So to find, we need to find a number,?, such that 0 0?. This is impossible, so we sa that 0 is undefined or does not exist. Special Slopes Slopes of Horizontal and Vertical Lines 0 Ex 8: Find the slope of the line in the graph x - Step One: Find two points on the line. (See graph.) -0 Step Two: Start at one of the points. Count how man units up or down ou would have to step to get to the other point. This is our rise. Note: If ou go down, the rise is negative; if ou go up, the rise is positive. Starting at the point on the right, we would have to step down 0 units, therefore the rise is 0. Step Three: Now determine how man units right or left ou would have to step to get to the other point. This is our run. Note: If ou go left, the run is negative; if ou go right, the run is positive. We would have to step right 4 units, therefore the run is 4. Step Four: Write the slope as a fraction, rise run. The slope is Linear Functions & Inequalities Part I page 8 of 4 /0/

9 Note: All horizontal lines have a slope of 0. If a car was driving from left to right, it would be going neither uphill nor downhill. Therefore, the slope is neither positive nor negative. Ex 9: Find the slope of the line passing through the points,6 and,. We will use the slope formula m x x, where, =,6 and,, x x. Substituting into the formula: 6 4 m, so the slope is undefined. 0 Note: The points in the example above will graph a vertical line. The slope of a vertical line is undefined. If a car tried to drive on a vertical line it would crash! Application of Slope: Rates of Change Ex 0: In 970, the price of a movie ticket at a particular theater was $.0. In 990 the price of a movie ticket at the same theater was $6.00. What is the rate of change of the cost of a movie ticket? Use correct units in our answer. Using the slope formula, we will find the change in the cost of a ticket over the change in time. change in cost ($) dollars per ear change in time (ears) QOD: Show algebraicall using the slope formula wh the slope of a horizontal line is 0 and the slope of a vertical line is undefined. Sample Exam Questions. Find the slope of the line. ANS: rise run Linear Functions & Inequalities Part I page 9 of 4 /0/

10 . Find the slope of the line. ANS: rise run 4 4. What is the slope of a line that goes through the points (, ) and (6, )? A. B. C. ANS: B D. 4. What is the slope of a line that goes through the points (, 0) and (7, 8)? A. 4 B. 4 C. 4 D. 4 ANS: D. Find the slope of the line. ANS: rise 0 0 run Linear Functions & Inequalities Part I page 0 of 4 /0/

11 6. Find the slope of the line. ANS: rise 4 undefined run 0 7. What is the slope of a line that goes through the points (6, ) and (6, )? A. B. C. 0 D. undefined ANS: D 8. What is the slope of a line that goes through the points (, ) and (4, )? A. B. C. 0 D. undefined ANS: C 9. Given the table of values below, determine whether or not the points all fall on a straight line. Explain our answer. x ANS: Yes, it is linear. The slope is constant from point to point and equals. Linear Functions & Inequalities Part I page of 4 /0/

12 0. Find the value of so that the line passing through the points (0,) and (4,) have a slope of. A. B. 9 C. 9 D. ANS: D. Use the table of values to find the rate of change and then explain its meaning. driving time (h) distance traveled (m) ANS: 8/, which means the car is traveling at a speed of 8 mph.. Use the table of values to find the rate of change and then explain its meaning. number of floor tiles x Area of tiled surface (in ) ANS: 6/, which means that each floor tile is 6 in.. When driving down a certain hill, ou descend feet for ever 000 feet ou drive forward. What is the slope of the road? ANS: The point (, 8) is on a line that has a slope of. Is the point (4, 7) on the same line? Explain our reasoning. ANS: Yes, the slope between the two points is also. Linear Functions & Inequalities Part I page of 4 /0/

13 . In 996, a compan had a profit of $,000,000. In 00, the profit was $86,000,000. If the profit increased the same amount each ear, find the average rate of change of the compan s profit in dollars per ear. ANS: $,00,000 per ear 6. Which statement is true about the characteristics of the linear functions in the graph below? (A) The are the same function. (B) The have the same slope. (C) The have the same x-intercepts. (D) The have the same -intercepts. ANS: C 7. Find the slope of the line in the graph. (A) (B) (C) (D) ANS: A Linear Functions & Inequalities Part I page of 4 /0/

14 8. Find the slope of the line that contains the points (7, 6) and (4, ). (A) (B) (C) (D) ANS: B 9. Find the slope of the line that contains the points (8, ) and (8, 8). (A) (B) 0 (C) 6 (D) undefined ANS: D 0. A job pas a base salar of $0,000. Each ear an emploee will earn an additional $,000. The graph of an emploee s salar over 0 ears is shown below. Emploee Salar over Time Annual Salar ($) Number of Years What would happen to the graph if the base salar was changed from $0,000 to $40,000? (A) The graph would translate down. (B) The graph would translate up. (C) The graph would rise less steepl from left to right. (D) The graph would rise more steepl from left to right. ANS: B Linear Functions & Inequalities Part I page 4 of 4 /0/

15 For questions and, use the table. x The ordered pairs (x, ) form a linear function. (A) True ANS: A (B) False. The value of changes b increasingl larger amounts for each change of in x. (A) True ANS: B (B) False Unit.4 To graph linear equations using slope intercept form. Coordinate Plane a plane formed b two real number lines (axes) that intersect at a right angle Horizontal Axis x-axis Vertical Axis -axis Ordered Pair the coordinates of a point x, 0 Quadrants the four sections of the plane formed b the axes Quadrant II Quadrant I The Cartesian (coordinate) Plane: -0-0 x Origin the point 0,0 - Quadrant III -0 Quadrant IV Linear Functions & Inequalities Part I page of 4 /0/

16 Plotting Points on the Coordinate Plane Ex : Plot the points A,, B 4,, and 0,6 C. 0 Note: Point A is in Quadrant I, B is in Quadrant II, and C is on the -axis x - -0 Solution of a Linear Equation in Two Variables an ordered pair, x,, that makes the equation true. Graph of a Linear Equations in Two Variables the set of all points, x,, that are solutions to the equation. The graph will be a line. Ex : Is the point 4, a solution to the linear equation x 0? Substitute the x- and -coordinates of the point into the equation true Because the values satisf the equation, the ordered pair is a solution. Graphing a Linear Equation To graph a linear equation, we will choose values for the independent variable, x, and substitute these values into the equation to find the corresponding values for the dependent variable,. It is helpful to organize the ordered pairs in a table (or t-chart ). Ex : Graph the equation x 4. Step One: Solve the equation for. x 4 x x x4 Linear Functions & Inequalities Part I page 6 of 4 /0/

17 Step Two: Choose at least values for x. Include 0, two positive integers, and two negative integers. x x 4 We will choose,,0,,. Step Three: Make a table and substitute each x-value into the equation to find the corresponding -value. Step Four: Plot each ordered pair from the table and connect to form a line x - -0 Ex 4: Graph the linear function 4 x. Step One: Solve the equation for. 4 x x Step Two: Choose at least values for x. Include 0, two positive integers, and two negative integers. Because we are multipling x b the fraction, we will choose x-values that are multiples of. 4,,0,,4 Teacher Note: Explain to students that it is easier to evaluate when we use x-values that are multiples of the denominator. Linear Functions & Inequalities Part I page 7 of 4 /0/

18 Step Three: Make a table and substitute each x-value into the equation to find the corresponding -value. Step Four: Plot each ordered pair from the table and connect to form a line x x x Ex : Graph the line. This equation is alread solved for, so we will simpl make a table. x 0 Note that the value is alwas, regardless of the value of x. Plotting the points and drawing the line, we see that the result is a horizontal line. - x - Ex 6: Graph the line x 4. This equation has no, and the x-coordinate must equal. Because is not restricted as part of the equation, we can choose an value for in our table. Plotting the points and drawing the line, we see that the result is a vertical line. x Note: This is the graph of a line, but it is not a function. - x Linear Functions & Inequalities Part I page 8 of 4 /0/ -

19 Equation of a Vertical Line: x a, where a is a real number Equation of a Horizontal Line: b, where b is a real number You Tr:. Use the linear equation x. Is the ordered pair, 4 a solution to the equation? Find five points that are solutions to the equation and graph the line.. Graph the lines x and. At what point do these two lines intersect? QOD: Wh isn t a vertical line a function? Activit: Graph the following lines on the graphing calculator. Write down the slope and -intercept of each line. Then, compare the steepness of the line to the parent function x. To graph on the calculator, enter the function into the Y= screen. Use a Zoom Standard window.. x Slope: m -intercept = 0. x Slope: m -intercept = steeper than x. 4. x 4 Slope: x Slope: m -intercept = 4 x is steeper m -intercept = x is steeper. 4x Slope: m 4 -intercept = steeper than x 6. x Slope: m -intercept = 0 steepness same as x Write down our conclusions: The coefficient of x is the slope, and the constant term is the - intercept. Linear Functions & Inequalities Part I page 9 of 4 /0/

20 Slope-Intercept Form of a Linear Equation: mx b, m slope, b = -intercept Ex 7: Write the equation of the line x 0in slope-intercept form. Identif the slope and -intercept. To write in slope-intercept form, solve for. x0 x The slope is, and the -intercept is. Graphing a Line in Slope-Intercept Form:. Plot the -intercept.. Starting at the -intercept, step to the next point on the line using the slope, rise run.. Begin at b, move m. Note: If the slope is POSITIVE ou ma rise up and run right, or ou ma rise down and run left. If the slope is NEGATIVE ou ma rise up and run left, or ou ma rise down and run right. Ex 8: Graph the line x. 4 0 Step One: Identif the slope and -intercept. Step Two: Plot a point at the -intercept. m -intercept = x - -0 Step Three: Step out the slope to plot another point. Do this a couple of times so that ou can draw an accurate line. Note: Because the slope is positive, we can rise up and run right 4, or rise down and run left 4. Draw the line connecting the points x - -0 Linear Functions & Inequalities Part I page 0 of 4 /0/

21 Ex 9: Graph the line f xx. Step One: Identif the slope and -intercept. Step Two: Plot a point at the -intercept. m -intercept = 0 Step Three: Step out the slope to plot another point. Do this a couple of times so that ou can draw an accurate line x - Note: Because the slope is negative, we can rise up and run left, or rise down and run right Draw the line connecting the points. Note: Sometimes ou must put the equation in slope-intercept form first x x Ex 0: Graph the equation of the line using slope-intercept form. Step One: Write the equation in slope-intercept form. x x Step Two: Identif the slope and -intercept. 0 m -intercept = Step Three: Graph the line b plotting the -intercept and then stepping out the slope x - -0 Linear Functions & Inequalities Part I page of 4 /0/

22 Ex : Graph the equation of the line 0.x using slope-intercept form. Step One: Write the equation in slope-intercept form. 0.x x Step Two: Identif the slope and -intercept. m -intercept = 0 Step Three: Graph the line b plotting the -intercept and then stepping out the slope x - -0 Ex : Graph the equation of the line x 48x using slope-intercept form. Step One: Write the equation in slope-intercept form. 6x 9 x 0 Step Two: Identif the slope and -intercept. m -intercept = -0-0 x - Step Three: Graph the line b plotting the -intercept and then stepping out the slope. -0 Ex : Find the slope and -intercept of the line. Describe the graph. Slope = 0, -intercept = The graph will be a horizontal line that intersects the -axis at. Linear Functions & Inequalities Part I page of 4 /0/

23 Application Problems Using Slope-Intercept Form Ex 4: Jack has $0 in his savings account, and plans to save $0 per week. Write an equation in slope-intercept form that represents the amount, A, Jack will have in his account after w weeks. Graph the equation in an appropriate window. Amount = 0 Number of Weeks + 0 A0w 0 A w On Your Own: What does the slope and -intercept represent on the graph in relation to the problem? You Tr: Graph the linear equation 8 x 6 using slope-intercept form. QOD: Can ou write the equation of a vertical line in slope-intercept form? Explain. Sample Exam Questions. Use the graph below. What is the equation of the line in the graph? (A) x (B) x (C) x (D) x Ans: D Linear Functions & Inequalities Part I page of 4 /0/

24 . What is the equation of the horizontal line through the point (4, 7)? (A) x = 4 (B) x = 7 (C) = 4 (D) = 7 Ans: D. When the function f = k + ac is graphed on the axes shown, what quantit corresponds to the intercept on the vertical axis? f (A) f (B) k (C) f k c (D) f k a Ans: B 4. A line is defined b the equation x. Which ordered pair does NOT represent a point on the line? (A) (, 0) (B) (0, ) (C) 7 (, ) (D) (, ) Ans: A. A certain child s weight was measured at 6.6 pounds. The child then gained weight at a rate pounds of 0.6 pounds per month. On a graph of weight versus time, what would 0.6 month represent? (A) (B) (C) The -intercept of the graph The x-intercept of the graph The slope of the graph Ans: C Linear Functions & Inequalities Part I page 4 of 4 /0/

25 Graphing Calculator Activit: Solving linear equations on the graphing calculator. Ex : Solve the equation 0.8x. graphicall on the calculator. Step One: Rewrite the equation in the form ax b x. 00.8x.7 Step Two: Graph the line 0.8x.7. Enter the line in the Y= screen.. Step Three: Find the zero using the CALC Menu. Kestrokes: Note: You ma enter in values for the bounds and the guess, or use the left and right arrows. Solution: x. Step Four: Check our answer on the home screen. Your calculator has the value of the zero stored as x. Alternate Method: Ex 6: Solve the equation x7 0x Because it is tedious to rewrite this equation in the form ax b 0 b hand, we will use an alternate method on the graphing calculator. Step One: Graph each side of the equation as two separate lines. x x 7 and 0 Step Two: Find the point of intersection. This is the solution to the equation x7 0x. Kestrokes: Linear Functions & Inequalities Part I page of 4 /0/

26 Note: The guess should be near the point of intersection. The x-coordinate is the solution. x Checking this b hand algebraicall, we have Application Problem x7 0x True 8 Ex 7: A small business makes and delivers box lunches. The calculate their average weekl cost C of delivering b lunches using the function C.b 7. Last week their cost was $600. How man lunches did the make last week? Solve algebraicall and graphicall. Because their cost was $600, we will substitute this value in for C. 600.b 7 Solve for b..b 0 b Solving graphicall, we will rewrite the equation as 0.b, graph the line.x, and find the zero. Solution : The business made 0 lunch boxes last week. You Tr: Solve the equation 96x graphicall and check our solution algebraicall. QOD: What are three different terms to describe the graphical solution to a linear equation in the form ax b 0? Unit. To use intercepts to graph linear functions in standard from x-intercept the x-coordinate of the point where the graph intersects the x-axis (Note: The - coordinate of the x-intercept is 0.) -Intercept the -coordinate of the point where the graph intersects the -axis (Note: The x- coordinate of the -intercept is 0.) Linear Functions & Inequalities Part I page 6 of 4 /0/

27 Ex : Find the x- and -intercept of the graph of the equation x 9. Finding the x-intercept: Let 0 and solve for x. Finding the -intercept. Let x 0 and solve for. x 0 9 x 9 9 x 09 9 Note: The line crosses the x-axis at the point 9,0 and the -axis at the point 0,. Sketching the Graph of a Line Using Intercepts Ex : Sketch the graph of the line x. To sketch a line, we onl need two points. We will use the intercepts as the two points. Step One: Find the x-intercept. Step Two: Find the -intercept. x 0 x x Step Three: Plot the intercepts on the axes and draw the line Ex : Sketch the graph of the line f x x. 4 To sketch a line, we onl need two points. We will use the intercepts as the two points. Step One: Find the x-intercept. 0 x 4 x 4 8 x x Linear Functions & Inequalities Part I page 7 of 4 /0/

28 Step Two: Find the -intercept. 0 4 Step Three: Plot the intercepts on the axes and draw the line. Note: Use a friendl scale to view the graph. Application Problem: Using a Linear Model Ex 4: Adult tickets to a football game cost $, and student tickets cost $. A school collects $7 at Frida night s game. Write an equation and draw a line that represents the possible number of adult tickets, a, and student tickets, s, that were sold at the game. Write the equation: as 7 Find the intercepts: Draw the graph: a 70 0 s 7 s a 0 7 a s Note: Ever whole-number point on the graph of the line are ordered pairs that represent the possible number of adults and students who purchased a ticket for the football game. You Tr: Find the x- and -intercepts of the graph of the equation 0.x 6. Graph the line. QOD: Describe the graph of a line that has no x-intercept. Write an equation of a line with no x- intercept. Linear Functions & Inequalities Part I page 8 of 4 /0/

29 Sample Exam Questions. What are the x- and -intercepts of 9x? (A) x-intercept = 9; -intercept = (B) x-intercept = ; -intercept = (C) x-intercept = ; -intercept = (D) x-intercept = ; -intercept = 9 Ans: B. Graph the equation: 6x. Ans: A. What are the intercepts of the line with equation x = 0? (A) ( 0, 0) and (0, ) (B) (6, 0) and (0, 6) (C) (, 0) and (0, 0) (D) (0, 0) and (0, 0) Ans: C Linear Functions & Inequalities Part I page 9 of 4 /0/

30 Unit. To graph linear inequalities in two variables. Linear Inequalit in Two Variables: an inequalit that can be written in one of the following forms ax b c, ax b c, ax b c, ax b c Solutions of a Linear Inequalit: the ordered pairs, x, that make the inequalit true Checking a Solution to a Linear Inequalit Ex : Is the ordered pair, a solution to the linear inequalit x 8? Substitute the ordered pair in for x and in the inequalit. Evaluate and determine if it makes the inequalit true. x False The ordered pair does not make the inequalit true, so it is NOT a solution. Half-Plane: one of the two planes that a line separates the coordinate plane into Graphing a Linear Inequalit. Graph the line ax b c. Use a dashed line for < or >, a solid line for or.. Choose a test point in one of the half-planes and substitute the ordered pair into the inequalit for x and. If the ordered pair makes the equation true, shade the halfplane that contains our test point. If it does not, shade the other half-plane. Ex : Graph the linear inequalit x. Step One: Graph the line x. 0 x-intercept =, -intercept = -0-0 x - -0 Note: We will use a solid line, because the inequalit is. Linear Functions & Inequalities Part I page 0 of 4 /0/

31 Step Two: Pick a test point in one of the half-planes and test it in the linear inequalit. We will choose 0,0. x 00 False Step Three: Because 0,0is NOT a solution to the inequalit, we will shade the other half-plane. Note: All of the solutions of the linear inequalit are all points on the line and in the shaded region x - -0 Ex : Graph the linear inequalit x. Step One: Graph the line x. Slope =, -intercept = 0 Note: We will use a dashed line, because the inequalit is <. Step Two: Pick a test point in one of the half-planes and test it in the linear inequalit. We will choose the point 0, x 0 True. 0 Step Three: Because 0, IS a solution to the inequalit, we will shade the half-plane that the point lies in. Ex 4: Sketch the graph of x in the coordinate plane. 0 Step One: Graph the line x. Note: We will use a solid line, because the inequalit is x - -0 Linear Functions & Inequalities Part I page of 4 /0/

32 Step Two: Pick a test point in one of the half-planes and test it in the linear inequalit. We will choose the point 0,0. x 0 False 0 Step Three: Because 0,0IS NOT a solution to the inequalit, we will shade the other half-plane x - Application Problem -0 Ex : A basketball team is 8 points behind with minutes left in the game. Write an inequalit that represents the number of -point and -point shots the team could score to earn at least 8 points. Graph the inequalit and give two different numbers of -point and -point shots the team could score. Assign labels: w = number of -point shots, r = number of -point shots Write an inequalit: wr 8 r 0 Graph the inequalit: w-intercept = 9, r-intercept = 6 Test point 0,0 : False 0 w Two possible - and -point combinations: Three -point shots and four -point shots (This corresponds to the ordered pair, 4, which lies on the line.) Six -point shots and three -point shots (This corresponds to the ordered pair 6,, which lies in the shaded region.) You Tr: Graph the linear inequalities.. 9. x 6 QOD: Explain how to determine if an ordered pair is a solution of a linear inequalit. Linear Functions & Inequalities Part I page of 4 /0/

33 Sample Exam Questions. Graph the linear inequalit x. Ans: B. Which is the graph of x <? (A) (B) x x (C) D) x x Ans: B Linear Functions & Inequalities Part I page of 4 /0/

34 . Use the graph. x Which inequalit is represented in the graph? (A) x (B) x (C) (D) Ans: D x For questions 4-6, use the inequalit. 4. (0, ) is a solution of the inequalit. (A) (B) True False. (, ) is a solution of the inequalit. (A) (B) True False 6. (, 0) is a solution of the inequalit. (A) (B) True False Ans: B, B, A Linear Functions & Inequalities Part I page 4 of 4 /0/

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