1. © A Very Good Teacher 2007 Exit Level TAKS Preparation Unit Objective 3

2. © A Very Good Teacher 2007 Interpreting Linear Functions Functions can be represented in different ways: y = 2x + 3 means the same thing as f(x) = 2x + 3 Linear Functions must have a slope (rate of change) and a y intercept (initial value). In a function…  the slope is the constant (number) next to the variable  the y intercept is the constant (number) by itself 3, Ac1A

3. © A Very Good Teacher 2007 Interpreting Linear Functions, cont… Example: Identify the situation that best represents the amount f(n) = 425 + 50n. Slope (rate of change) = Y intercept (initial value) = 50425 Find an answer that has: 425 as a non-changing value and 50 as a recurring charge every month, every year, etc… Something like Joe has $425 in his savings account and he adds $50 every month. 3, Ac1A

4. © A Very Good Teacher 2007 Converting Tables to Equations When given a table of values, USE STAT! Example: What equation describes the relationship between the total cost, c, and the number of books, b? bc 1075 15100 20125 25150 Answer: c = 5x + 25 3, Ac1C

5. © A Very Good Teacher 2007 Converting Graphs to Equations Make a table of values Then, use STAT! Example: Which linear function describes the graph shown below? 3, Ac1C xy Answer: y = -.5x + 4 -2 5 0 4 2 3 4 2

6. © A Very Good Teacher 2007 Converting Equation to Graph Graph the function in y = Example: Which graph best describes the function y = -3.25x + 4? Find an answer that has the same y intercept and x intercept as the calculator graph. 3, Ac1C

7. © A Very Good Teacher 2007 Equations that are in Standard Form Sometimes your equations won’t be in y = mx + b form. They will be in standard form: Ax + By = C You must convert them to use the calculator! Example: 3x + 2y = 12 Step 1: Move the x -3x 2y = -3x + 12 Step 2: Divide everything by the number in front of y 222 3, Ac1C

8. © A Very Good Teacher 2007 Slope and Rate of Change (m) Slope and rate of change are the same thing! They both indicate the steepness of a line. Three ways to find the slope of a line: By Formula:By Counting:By Looking: You must have 2 points on a line You must have a graph You must have an equation 3, Ac2A

9. © A Very Good Teacher 2007 Slope and Rate of Change (m), cont… By Formula: Find two points on the graph (they won’t be given to you) 3, Ac2A (0, 4) and (2, 3)

10. © A Very Good Teacher 2007 Slope and Rate of Change (m), cont… By Counting Find two points on the graph 3, Ac2A Down 2 Right 4

11. © A Very Good Teacher 2007 Slope and Rate of Change (m), cont… By Looking The equation won’t be in y = mx + b form You’ll have to change it If in Standard Form use Process on Slide 7 If in some other form, you’ll have to work it out… 3, Ac2A Example: What is the rate of change of the function 4y = -2(x – 24)? Try to get rid of any parentheses and get the y by itself (isolated). 4y = -2x + 24 444

12. © A Very Good Teacher 2007 Slope and Rate of Change (m), cont… Special Cases Horizontal lines line y = 4 3, Ac2A Vertical lines like x = 4 Have slope of zero, m = 0 Have slope that is undefined

13. © A Very Good Teacher 2007 m and b in a Linear Function Changes to m, the slope, of a line effect its steepness 3, Ac2C Changes to b, the y intercept, of a line effect its vertical position (up or down) y = 1x + 0 y = 3x + 0 y = 1/3 x + 0 y = 1x + 0 y = 1x + 3 y = 1x - 4

14. © A Very Good Teacher 2007 m and b in a Linear Function, cont… Parallel Lines have equal slope (m) y = ¼ x – 3 and y = ¼ x + 6 Perpendicular Lines have opposite reciprocal slope (m) y = ¼ x – 5 and y = -4x + 15 Lines with the same y intercept will have the same number for b y = ¾ x – 9 and y = 5x – 9 3, Ac2C

15. © A Very Good Teacher 2007 Linear Equations from Points Make a table USE STAT Example: Which equation represents the line that passes through the points (3, -1) and (-3, -3)? xy 3 -3 Answer: 3, Ac2D

16. © A Very Good Teacher 2007 Intercepts of Lines To find the intercepts from a graph… just look ! The x intercept is where a line crosses the x axis The y intercept is where a line crosses the y axis 3, Ac2E (4, 0) (0, 2)

17. © A Very Good Teacher 2007 Intercepts of Lines, cont… To find intercepts from equations, use your calculator to graph them Example: Find the x and y intercepts of 4x – 3y = 12. -4x -3y = -4x + 12 -3 x intercept: (3, 0) y intercept: (0, -4) 3, Ac2E

18. © A Very Good Teacher 2007 Direct Variation Set up a proportion! Make sure that similar numbers appear in the same location in the proportion Example: If y varies directly with x and y is 16 when x is 5 what is the value of x when y = 8? 3, Ac2F 16x = 5(8) 16x = 40 16 x = 2.5

19. © A Very Good Teacher 2007 Direct Variation, cont… To find the constant of variation use a linear function (y = kx) and find the slope The slope, m, is the same thing as k Example: If y varies directly with x and y = 6 when x = 2, what is the constant of variation? y = kx 6 = k(2) 22 3 = k The equation for this situation would be y = 3x 3, Ac2F