Embed Size (px)
Citation preview
Reporting Strand 4: Graphs of Functions
CCSS Instructional Focus Understand Solutions (A.REI.10) Understand that all solutions to an equation in two variables are contained on the
graph of that equation. Graph Functions (F.IF.7b) Graph a variety of functions expressed symbolically, including all of the following:
linear, piecewise, absolute value, and step
1. The following equation represents a linear equation: 𝑦 = 2𝑥 – 4 Sketch the linear equation using an x-‐ & y-‐table. An x-‐ & y-‐table is a method that you can use to graph any type of equation that is new to you; however, once we learn a method of graphing you may not use an x-‐ & y-‐table.
LINEAR EQUATIONS
SLOPE
Y-‐INTERCEPT
SLOPE-‐INTERCEPT FORM
POINT-‐SLOPE FORM
STANDARD FORM
SLOPE-‐REFERENCE GUIDE
Find the slope of the line that passes through each pair of points. 2. (0, 0), (3, 3) 3. (4, 4), (5, 3) 4. (−3, 4), (4, 4) 5. (−2,−4), (−2, 3)
𝑦 − 𝑦! = 𝑚(𝑥 − 𝑥!) (point-‐slope form)
𝑦 − 3 = 5(𝑥 + 4)
3𝑥 − 2𝑦 = 4 (standard form)
6. Find the slope given the following lines.
7. Find the slope from the following table of value. x y 1 8 4 2 5 0 10 -‐10
Find the slope and y-‐intercepts for the following equations. 8. 𝑦 = 6𝑥 − 4 9. 𝑦 − 2 = −3𝑥 10. 4𝑥 + 2𝑦 = 6
11a. Which of the following points lie on the line of the graph shown to the right? a. (3, 8) b. (-‐1, 2) c. (15, 20) d. (-‐6, -‐1) e. (-‐13, -‐7)
11b. How many points would satisfy the equation of the line shown? 11c. What are all the solutions to the equation of the line? Given the following linear equations, identify which solutions would lie on the line of the equations. 12. 𝑦 = − !
!𝑥 − 6 13. 𝑦 − 3 = 2(𝑥 + 4)
a. (2, 7) a. (3, 17) b. (4, -‐8) b. (-‐5, 1) c. (-‐2, -‐5) c. (-‐0.5, 10) d. (32, -‐22) d. (5, 17) e. (6, 3) e. (-‐5, -‐3)
How many points would satisfy the equations in #12-‐13? What are all the solutions to the equations of these lines? Each pair of points lies on a line with the given slope. Find x or y.
14. 4, 3 , 5, 𝑦 ; 𝑠𝑙𝑜𝑝𝑒 = 3
15. 3, 𝑦 , 1, 9 ; 𝑠𝑙𝑜𝑝𝑒 = − !
!
16. (3, 5), (𝑥, 2); 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑠𝑙𝑜𝑝𝑒
Graph the following equations. 17. 𝑦 = − !
!𝑥 − 4
18. Graph the equation: −2(3𝑥 + 4) + 𝑦 = 0
19. 𝑦 − 4 = 3 𝑥 + 2 20. 𝑦 − 1 = !! (𝑥 − 2)
21. 𝑦 = !
! 𝑥 + 2 22. 6𝑥 + 3𝑦 = 9
Write the equation in slope-‐intercept form of the line that passes through the given points. Then find the x-‐ and y-‐intercepts of the line. 23. −2,−1 𝑎𝑛𝑑 4, 2 24. −2, 4 , 𝑎𝑛𝑑 3,−1 25. (−6, 5) 𝑎𝑛𝑑 (1, 0)
Write the equation of the line in point-‐slope form. Then transform the equation into slope-‐intercept form. 26. 3,−8 ; 𝑚 = − !
! 27. (−1,−2), (2, 4) 28. (−6, 6), (3, 3)
29. Evaluate the function for the given value of x.
𝑓 𝑥 =3, 𝑖𝑓 𝑥 ≤ 02, 𝑖𝑓 𝑥 > 0 𝑔 𝑥 =
𝑥 + 5, 𝑖𝑓 𝑥 ≤ 32𝑥 − 1, 𝑖𝑓 𝑥 > 3 ℎ 𝑥 =
!!𝑥 − 4, 𝑖𝑓 𝑥 ≤ −23 − 2𝑥, 𝑖𝑓 𝑥 > −2
𝑓 2 = 𝑓 −4 = 𝑓 0 = 𝑓 !
!=
𝑔 7 = 𝑔 0 = 𝑔 −1 = 𝑔 3 = ℎ −4 = ℎ −2 = ℎ −1 = ℎ 6 = 30. Evaluate the function for the given value of x.
𝑓 𝑥 = 3𝑥 − 5, 𝑥 > 4𝑥!, 𝑥 ≤ 4
𝑓 7 = 𝑓 4 = 𝑓 −3 = 31. Evaluate the function for the given value of x.
𝑓 𝑥 =−2 𝑥 + 1 , 𝑥 ≤ 13, 1 < 𝑥 < 36 − 2𝑥, 𝑥 ≥ 3
𝑓 10 = 𝑓 2 = 𝑓 0 =
PIECEWISE LINEAR EQUATIONS
Graph the following piecewise linear functions.
32. 𝑓 𝑥 =𝑥 + 3, 𝑖𝑓 𝑥 ≤ 02𝑥, 𝑖𝑓 𝑥 > 0 33. 𝑓 𝑥 = −2, 𝑥 < 0
3, 𝑥 ≥ 0
34. 𝑔 𝑥 = −𝑥 + 2, 𝑥 < 2𝑥 − 2, 𝑥 ≥ 2 35. 𝑓 𝑥 =
4𝑥 − 2, 𝑥 ≥ 2− !
!+ 4, 𝑥 < 2
36. 𝑓 𝑥 = 𝑥 + 5, 𝑥 < −2−2𝑥 − 1, 𝑥 ≥ −2 37. 𝑓 𝑥 =
2𝑥 + 1, 𝑥 ≥ 1!!𝑥 − 3, 𝑥 < 1
We will graph the absolute value equation of 𝑦 = |𝑥| to see the general shape of absolute value functions.
x y -‐3 -‐2 -‐1 0 1 2 3
ABSOLUTE VALUE EQUATIONS
Graph the following absolute value functions. 38. 𝑓 𝑥 = !
!𝑥 − 2 + 5 39. 𝑓 𝑥 = |𝑥 + 3|
40. 𝑓 𝑥 = −2 𝑥 − 1 41. 𝑓 𝑥 = 3 𝑥 − !
!
41. 𝑓 𝑥 = − 𝑥 − 3 − 4
STEP EQUATIONS Floor functions Ceiling Functions
(round down always) (round up always)
𝑓 𝑥 = 𝑥 𝑓 𝑥 = 𝑥
GENERAL TIPS
42. 𝑔 𝑥 = 𝑥 + 4 43. 𝑓 𝑥 = 𝑥 + 2
44. ℎ 𝑥 = 3 𝑥 45. 𝑓 𝑥 = !
!𝑥
46. 𝑦 = 2𝑥 47. 𝑦 = − !
!𝑥
PARENT FUNCTIONS Linear Absolute Value
Square Root Quadratic
Cubic
TRANSFORMATIONS
Transformation Appearance in Function Vertical Translation Up 𝑓 𝑥 + 𝑘
Vertical Translation Down 𝑓 𝑥 − 𝑘 Horizontal Translation Left 𝑓(𝑥 + ℎ) Horizontal Translation Right 𝑓(𝑥 − ℎ) Stretch/Compress Vertically 𝑎𝑓 𝑥
Reflection in x-‐axis −𝑓(𝑥) Reflection in y-‐axis 𝑓(−𝑥)
Describe the transformation that occurred. 48. 𝑦 = −2 𝑥 + 3 ! + 5 49. 𝑦 = 𝑥! − 1 50. 𝑓 𝑥 = 2 𝑥 − 1 51. ℎ 𝑥 = 𝑥 − 2 52.𝑔 𝑥 = 𝑥! + 3 53. 𝑓 𝑥 = −𝑥 + 5 − 2 54. 𝑔 𝑥 = 3 𝑥 55. 𝑦 = −𝑥! + 1 56. 𝑦 = 𝑥 − 4 Given the parent function and a description of the transformation(s), write the equation of the transformed function. 57. Absolute value – Vertical shift up 5, horizontal shift right 3 58. Square root – vertical compression (stretch) by !
!
59. Cubic – reflected over the x-‐axis and vertical shift down 2 60. Quadratic – horizontal shift left 8 & reflected over the y-‐axis 61. Linear – vertical stretch by 5, vertical shift down 3
Graph the following functions using the parent graph. 62. 𝑦 = 3(𝑥 + 1)! 63. 𝑦 = 2 𝑥 + 2 − 3
64. 𝑦 = − 𝑥 − 5 65. 𝑦 = 𝑥! + 4
