Linear Functions: Identifying, Writing, and Graphing

1. Do Now 11/7/18 • Take out classwork/HW. • “Take-Home” Quiz • Copy HW in your planner. • Text p. 169, #1-10 all – TEXTBOOK only • Chapter 3 Test Monday • Search your computer or re-download the “How Are You Doing?” Chapter 3 Graphing Linear Functions worksheet from Schoology. Submit the assignment when it is completed.

2. Learning Goal • SWBAT identify, write, graph, and transform linear functions Learning Target • SWBAT review identifying, writing, graphing, and transforming linear functions

3. Section 3.1 “Functions”

4. Set 1 Set 2 RELATION Domain Range Input Output X-coordinate Y-coordinate Independent Dependent -can be ANY value in the domain -DEPENDS on the input values -”y varies depending on the value of x”

5. 2468 1234 RELATION MAPPING DIAGRAM GRAPH Input Output

6. Is a relation that pairs each input with EXACTLY ONE output. FUNCTION-

7. 2468 0101 35 5123 Is a relation that pairs each input with EXACTLY ONE output. FUNCTION- FUNCTION NOT A FUNCTION Each input must be paired with only ONE output

8. Section 3.2 “Linear Functions” LINEAR FUNCTION- -has a constant rate of change (slope) and can be represented by a linear equation in two variables. A linear equation in two variables,x and y, is an equation that can be written in the form y = mx + b, where, m and b, are constants.

9. -has a constant rate of change (slope) and its graph is a LINE. -does not have a constant rate of change (slope) and therefore, its graph is NOT a line. LINEAR FUNCTION- NONLINEAR FUNCTION-

10. Identifying Linear Functions Using Equations A linear equation in two variables, x and y, is a LINEAR FUNCTION if it can be written in the form y = mx + b Yes, the equation can be written in the form y = mx + b No, the equation cannot be written in the form y = mx + b No, the equation cannot be written in the form y = mx + b Yes, the equation can be written in the form y = mx + b

11. Graphs of Linear Functions The domain of the function depends on the real- life context of the function -set of input values that consists of all numbers in an interval -set of input values that consists of only certain numbers in an interval DISCRETE- CONTINUOUS-

12. Section 3.3 “Function Notation” Function Notation- a linear function written in the form y = mx + b where y is written as a function f. x-coordinate f(x) = mx + b This is read as ‘f of x’ slope y-intercept f(x) is another name for y. It means “the value of f at x.” g(x) or h(x) can also be used to name functions

13. On Your Own For find the value of x for which g(x) = -4. Evaluate f(x) = 2x – 7 when x = 4? f(4) = 2(4) – 7 (-4) = 1/3x – 2 Simplify -2 = 1/3x f(4) = 8 – 7 f(4) = 1 -6 = x When x = 4, f(x) = 1 When x = -6, g(x) = -4

14. Interpreting Function Notation

15. Domain and Range • Domain = values of ‘x’ for which the function is defined. • Range = the values of f(x) where ‘x’ is in the domain of the function f. • The graph of a function f is the set of all points (x, f(x)).

16. Graphing a Function Graph the function y = 4 – 2x with the domain -2, -1, 0, 1, and 2. Then identify the range of the function. y-axis 10 (-2,8) 8 (-1,6) 6 4 (0,4) -2 8 (1,2) 2 -1 6 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 0 4 x-axis (2,0) -2 1 2 -4 2 0 -6 -8 The range of the function is discrete and is 0, 2, 4, 6, 8. -10

17. Graph an Equation Graph the equation y = 2 HORIZONTAL LINE y = b STEP2 STEP3 STEP1 Solve the equation for y. Make a table by choosing a few values for x and then finding values for y. Plot the points. Notice the points appear on a line. Connect the points drawing a line through them.

18. Graph an Equation Graph the equation x = -1. VERTICAL LINE x = a STEP2 STEP3 STEP1 Solve the equation for x. Make a table by choosing a few values for x and then finding values for y. Plot the points. Notice the points appear on a line. Connect the points drawing a line through them.

19. the x-coordinate of the point where the line crosses the x-axis x-intercept- To find the x-intercept, solve for ‘x’ when ‘y = 0.’ Find the x-intercept of the graph 2x + 3y = 12. 2x + 3y = 12 2x + 3(0)= 12 2x = 12 x = 6

20. y-intercept- the y-coordinate of the point where the line crosses the y-axis To find the y-intercept, solve for ‘y’ when ‘x = 0.’ Find the y-intercept of the graph 2x + 3y = 12. 2x + 3y = 12 2(0)+ 3y= 12 3y = 12 y = 4

21. Graph an Equation Using Intercepts Graph the equation 2x + 3y = 12. y-axis 10 Plot the two intercepts as points and then connect the points using a line. 8 6 4 The x-intercept = 6 (0,4) 2 (6,0) (6,0) -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 x-axis The y-intercept = 4 -2 (0,4) -4 -6 -8 -10

22. Slope Review The slope m of a line passing through two points and is the ratio of the rise change to the run. y run rise x

23. Section 3.5 “Graphing Equations in Slope-Intercept Form” SLOPE-INTERCEPT FORM- a linear equation written in the form y-coordinate x-coordinate y = mx + b slope y-intercept

24. y = 3x + 4 y = -3x + 2 y = 2 – 3x 3x +y = 2 The line has a slope of – 3 and a y-intercept of 2. The equation is in the formy =m x +b. So, the slope of the line is 3, and they-intercept is4. Rewrite the equation in slope-intercept form by solving for y. Identifying Slope and the Y-Intercept Slope of 3 means: y = mx + b Is this in slope-intercept form?

25. Graph an Equation Using the Slope-Intercept Form Graph the equation y = 2 - 2x. y-axis 5 4 Rewrite in slope-intercept form 3 2 1 slope y-intercept -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 x-axis -1 Slope of -2 means: -2 OR -3 Slope of -2 means: -4 -5

26. Transformation changes the size, shape, position, or orientation of a graph Translation – slides a graph horizontally or vertically Reflection – flips a graph over the x-axis or y-axis Stretches – changes slope of the line and moves graph farther away from axis Shrinks – changes slope of line and moves graph closer to the axis

27. Translations “Vertical” “Horizontal” y = f(x) + k y = f(x - h) Reflections “Over the x-axis” “Over the y-axis” y = -f(x) y = f(-x) Stretches “Horizontal” “Vertical” y = f(1/2x) y = 2f(x) Shrinks “Horizontal” “Vertical” y = f(2x) y =1/2f(x)

28. Transformations of Graphs Output Transformation Input Transformation y = af(x-h) + k or y = f(ax-h) + k If a is negative If a is negative Step 1: Translate the graph of y = f(x) horizontally h units. Step 2: Use ‘a’ to stretch or shrink the graph. Step 3: Reflect the resulting graph when a < 0. Step 4: Translate the resulting graph vertically k units.

29. Section 3.7 “Graphing Absolute Value Functions” Absolute Value Function- a function that contains an absolute value expression. The graph is a ‘V’ shape and symmetric about the y-axis.

30. Absolute Value in Vertex Form | -h| f(x) = a |x + k If a is negative Translates the graph horizontally h units. Stretches or shrinks the graph by a factor of ‘a’. Reflects the graph over the x-axis when a < 0. Translates the graph vertically k units. When the graph is in vertex form, the vertex of the graph of f(x) is (h, k).

32. “Chapter 3 Review – Scavenger Hunt” Complete exercises #2-11 all, 14-17 all, 22-36 evens on page 164 in your textbook. Use the PowerPoints (Sections 3.1 – 3.7) on the website as a review and reference. With your 7:00 partner.

33. Homework • Text p. 169, #1-10 all • Chapter 3 Test Monday