Identifying Linear Functions You can determine if a function is linear by its graph, ordered pairs, or equation. Identify whether the graph represents a linear function. Step 1: Determine whether the graph is a function. Every x-value is paired with exactly one y-value; therefore, the graph is a function. Continue to step 2. Step 2: Determine whether the graph is a straight line. Conclusion: Because this graph is a function and a straight line, this graph represents a linear function. Identify whether {(4, 3), (6, 4), (8, 6)} represents a linear function. Step 1: Write the ordered pairs in a table. Step 2: Find the amount of change in each variable. Determine if the amounts are constant. Conclusion: Although the x-values show a constant change, the y-values do not. Therefore, this set of ordered pairs does not represent a linear function. Identify whether the function y 5x 2 is a linear function. Try to write the equation in standard form (Ax By C). y 5x 2 5x 5x 5x y 2 In standard form, x and y have exponents of 1 are not multiplied together are not in denominators, exponents, or radical signs Conclusion: Because the function can be written in standard form, (A 5, B 1, C 2), the function is a linear function. Tell whether each graph, set of ordered pairs, or equation represents a linear function. Write yes or no. 1. 2. 3. x y 9 5 5 10 1 15 4. {( 3, 5), ( 2, 8), ( 1, 12)} 5. 2y 3x 2 6. y 4x 7
Give the domain and range for the graphs below. 7. 8. 9. 10. Tyler makes $10 per hour at his job. The function f(x) 10x gives the amount of money Tyler makes after x hours. Graph this function and give its domain and range. Using Intercepts continued You can find the x- and y-intercepts from an equation. Then you can use the intercepts to graph the equation. Find the x- and y-intercepts of 4x 2y 8. To find the x-intercept, substitute 0 for y. To find the y-intercept, substitute 0 for x. 4x 2x 8 4x 2(0) 8 4x 8 4x 4 8 4 x 2 4x 2y 8 4(0) 2y 8 2y 8 2y 2 8 2 y 4 The x-intercept is 2. The y-intercept is 4. Use the intercepts to graph the line described by 4x 2y 8. Because the x-intercept is 2, the point (2, 0) is on the graph. Because the y-intercept is 4, the point (0, 4) is on the graph. Plot (2, 0) and (0, 4).
Draw a line through both points. Find the x- and y-intercepts. 1. 2. 3. 4. The volleyball team is traveling to a game 120 miles away. Their average speed is 40 mi/h. The graphed line describes the distance left to travel at any time during the trip. Find the intercepts. What does each intercept represent?
Use intercepts to graph the line described by each equation. 5. 3x 9y 9 6. 4x 6y 12 7. 2x y 4 Rate of Change and Slope A rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. The table shows the average retail price of peanut butter from 1986 to 1997. Find the rate of change in cost for each time interval. During which time interval did the cost increase at the greatest rate? Year 1986 1987 1989 1992 1997 Cost per lb ($) 1.60 1.80 1.81 1.94 1.78 Step 1: Identify independent and dependent variables. Year is independent. Cost is dependent. Step 2: Find the rates of change. 1986 to 1987 1987 to 1989 change in cost 1.80 1.60 change in years 1987 1986 0.20 1 0.2 change in cost change in years 1.81 1.80 1989 1987 0.01 2 0.005 1989 to 1992 change in cost 1.94 1.81 0.13 0.043 change in years 1992 1989 3 1992 to 1997 change in cost 1.78 1.94 change in years 1997 1992 0.16 0.032 5 The cost increased at the greatest rate from 1986 to 1987. greatest rate of change This rate of change is negative. The price went down during this time period. The table shows the average retail price of cherries from 1986 to 1991. Find the rate of change in cost for each time interval. Year 1986 1988 1989 1991 Cost per lb ($) 1.27 1.63 1.15 2.26
1. 1986 to 1988 change in cost change in years 2. 1988 to 1989 change in cost change in years 3. 1989 to 1991 change in cost change in years 4. Which time interval showed the greatest rate of change? 5. Was the rate of change ever negative? If so, when?
Find the slope of each line. 6. 7. 8. 9. 10. 11. The Slope Formula You can find the slope of a line from any two ordered pairs. The ordered pairs can be given to you, or you might need to read them from a table or graph. Find the slope of the line that contains ( 1, 3) and (2, 0). Step 1: Name the ordered pairs. (It does not matter which is first and which is second.) first ordered pair ( 1, 3) (2, 0) second ordered pair Step 2: Label each number in the ordered pairs. ( 1, 3) (2, 0) (x 1, y 1) (x 2, y 2) Step 3: Substitute the ordered pairs into the slope formula. y2 y1 m x x 2 1 0 3 2 ( 1)
3 3 1 The slope of the line that contains ( 1, 3) and (2, 0) is 1. Find the slope of each linear relationship. 1. 2. x y 3. The line contains (5, 2) and (7, 6). 4 5 8 3 12 1 16 1
Find the slope of the line described by each equation. 4. 2x 5y 10 5. 4x 2y 8 6. 6x 2y 12 7. 8y 4x 32 8. 6y 8x 24 9. 1 x 2y 3 2 Direct Variation A direct variation is a special type of linear relationship. It can be written in the form y kx where k is a nonzero constant called the constant of variation. You can identify direct variations from equations or from ordered pairs. Tell whether 2x 4y 0 is a direct variation. If so, identify the constant of variation. First, put the equation in the form y kx. 2x 4y 0 2x 2x Add 2x to each side. 4y 2x 4y 4 2x 4 y 1 2 x Divide both sides by 4. Because the equation can be written in the form y kx, it is a direct variation. Tell whether the relationship is a direct variation. If so, identify the constant of variation. x 2 4 6 y 1 2 3 If we solve y kx for k, we get: y kx k Find k for each ordered pair. This means find for each ordered pair. If they are the same, the relationship is a direct variation. The constant of variation is 1 2. This is a direct variation. Tell whether each equation or relationship is a direct variation. If so, identify the constant of variation. 1. x y 7 2. 4x 3y 0 3. 8y 24x
4. x 4 2 10 y 2 1 5 5. x 5 12 8 y 17.5 42 28 6. x 6 8 10 y 8 10 12
7. The value of y varies directly with x, and y 8 when x 2. Find y when x 10. 8. The value of y varies directly with x, and y 5 when x 20. Find y when x 35. 9. The cost of electricity to run a personal computer is about $2.13 per day. Write a direct variation equation for the electrical cost y of running a computer each day x. Then graph. Slope-Intercept Form An equation is in slope-intercept form if it is written as: y mx b. m is the slope. b is the y-intercept. A line has a slope of 4 and a y-intercept of 3. Write the equation in slope-intercept form. y mx b Substitute the given values for m and b. y 4x 3 A line has a slope of 2. The ordered pair (3, 1) is on the line. Write the equation in slope-intercept form. Step 1: Find the y-intercept. y mx b y 2x b Substitute the given value for m. 1 2( 3) b Substitute the given values for x and y. 1 6 b Solve for b. 6 5 b 6 Step 2: Write the equation. y mx b y 2x 5 Substitute the given value for m and the value you found for b. Write the equation that describes each line in slope-intercept form. 1. slope 1, y-intercept 3 4 2. slope 5, y-intercept 0
3. slope 7, y-intercept 2 4. slope is 3, ( 4, 6) is on the line. 5. slope is 1, ( 2, 8) is on the line. 2 6. slope is 1, (5, 2) is on the line.
Write the following equations in slope-intercept form. 7. 5x y 30 8. x y 7 9. 4x 3y 12 10. Write 2x y 3 in slope-intercept form. Then graph the line. Point-Slope Form You can graph a line if you know the slope and any point on the line. Graph the line with slope 2 that contains the point (3, 1). Step 1: Plot (3, 1). Step 2: The slope is 2 or 2 ; Count 2 up and 1 1 right and plot another point. Step 3: Draw a line connecting the points. Graph the line with the given slope that contains the given point. 1. slope 2 3 1 ; ( 3, 3) 2. slope ; ( 2, 4) 3. slope 3; ( 2, 2) 2 4. slope 3 2 ; (1, 2) 5. slope 2; ( 3, 2) 6. slope 2 ; (2, 4) 3
Point-Slope Form continued You can write a linear equation in slope-intercept form if you are given the slope and a point on the line, or if you are given any two points on the line. Write an equation that describes each line in slope intercept form. slope = 3, (4, 2) is on the line Step 1: Write the equation in point-slope form. (10, 1) and (8, 5) are on the line Step 1: Find the slope. y2 y1 5 1 4 y 2 = 3(x 4) m 2 x x 8 10 2 Step 2: Write the equation in slope-intercept form by solving for x 2 1 Step 2: Substitute the slope and one point into the point-slope form. Then write in slope-intercept form. y 2 = 3(x 4) y y1 m( x x1) y 2 = 3x 12 y 5 2 x 8 2 2 y 5 2x 16 y = 3x 10 5 5 y = 2x + 21 Write the equation that describes the line in slope-intercept form. 7. slope 3; (1, 2) is on the line 8. slope 1 4 ; (8, 3) is on the line 9. slope 4; (2, 8) is on the line 10. (1, 2) and (3, 12) are on the line 11. (6, 2) and ( 2, 2) are on the line 12. (4, 1) and (1, 4) are on the line Slopes of Parallel and Perpendicular Lines Two lines are parallel if they lie in the same Two lines are perpendicular if they intersect to form right angles. Identify which lines are perpendicular. If the product of the slopes of two lines is 1, the two lines are perpendicular.
plane and have no points in common. The lines will never intersect. Identify which lines are parallel. y 2x 4; y 3x 4; y 2x 1 If lines have the same slope, but different y-intercepts, they are parallel lines. y 2x 4; y 3x 4; y 2x 1 m 2, m 3 m 2 b 4 b 4 b 1 y 2x 4 and y 2x 1 are parallel. Identify which two lines are parallel. Then graph the parallel lines. 1. y 4x 2; y 2x 1; y 2x 3 Identify which two lines are perpendicular. Then graph the perpendicular lines. 2. y 2 3 x 2; y 3 2 x 1; y 2 3 x 3
Slopes of Parallel and Perpendicular Lines continued Write an equation in slope-intercept form for the line that passes through (2, 4) and is parallel to y 3x 2. Step 1: Find the slope of the line. The slope is 3. Step 2: Write the equation in point-slope form. y y 1 m(x x 1) y 4 3(x 2) Step 3: Write the equation in slope-intercept form. y 4 3(x 2) y 4 3x 6 4 4 y 3x 2 Write an equation in slope-intercept form for the line that passes through (2, 5) and is perpendicular to y 2 3 x 2. Step 1: Find the slope of the line and the slope for the perpendicular line. The slope is 2. The slope of the 3 perpendicular line will be 3 2. Step 2: Write the equation (with the new slope) in point-slope form. y y 1 m(x x 1) y 5 3 (x 2) 2 Step 3: Write the equation in slope-intercept form. y 5 3 (x 2) 2 y 5 3 2 x 3 5 5 y 3 2 x 8 Write the slope of a line that is parallel to, and perpendicular to, the given line. 3. y 6x 3 parallel: perpendicular: 4. y 4 x 1 parallel: perpendicular: 3 5. Write an equation in slope-intercept form for the line that passes through (6, 5) and is parallel to y x 4. 6. Write an equation in slope-intercept form for the line that passes through (8, 1) and is perpendicular to y 4x 7.