Lesson 2.2.3

HW: 2-125 to 2-131, any 5

Learning Target: Scholars investigate one more transformation, f(−x), the reflection of f(x) across the y-axis. Then they will compare f(−x) and −f(x), for a variety of parent graphs and develop the definitions for even and odd functions. In your Geometry course you transformed figures just like you transformed parent graphs in this course. Today you will look more at geometric transformations and you will explore what happens when you take the opposite of x before applying the operations of the function. That is, you will investigate f(–x).

2-121. In Geometry, you called the transformation of figures “translations,” “reflections,” “rotations,” and “dilations.” Refer to your Parent Graph Toolkit from problem 2-104 and/or your Learning Log entry in problem 2-106 as you complete parts (a) through (d) below.

  1. What kind of a geometric transformation have you made when you replace f (x) with f (x) + k? Be as specific as you can. Explore using the2-121(a) eTool(Desmos).
  2. What kind of geometric transformation occurs when you replace f (x) with–f (x)? Be as specific as you can. Explore using the2-121(b) eTool(Desmos).
  3. What kind of transformation is f (x –h)? Explore using the2-121(c) eTool(Desmos).
  4. What kind of transformation is a· f (x)? Be specific. Explore using the 2-121(d) eTool(Desmos).

2-122. Investigate the transformation y =f (–x)as directed below.

  1. For each of the parent graphs you have investigated so far, investigate what happens to the graph when you replace x with –x. For each parent function, draw the original and the new graph on the same set of axes in different colors. Explore using the2-122 eTool(Desmos).
  2. For each parent equation, substitute –x for x and algebraically simplify the result.
  3. Describe the geometric transformation that occurs when you replace f(x) with f(–x).

2-123.Functions can be categorized as even functions orodd functions. With your team sort the functions you investigated in problem 2-122 into the following three groups:

 EVEN FUNCTIONS: All functions wheref(–x) = f (x).

 ODD FUNCTIONS: All functions wheref(–x) =–f(x).

 FUNCTIONS THAT ARE NEITHER EVEN nor ODD.

2-125.Decide whether each of the following functions is even, odd, or neither. Show or explain your reasoning.

  1. y = (x + 2)2

2-126.For each of the following functions sketch the graph of the original and ofy=f(–x). 2-126 HW eTool (Desmos).

  1. .
  2. Is either of these functions odd or even? Justify your answer.

2-127.A parabola has vertex (2, 3) and contains the point (0, 0). Find an equation that represents this parabola.

2-128.For each equation below, find the x- and y-intercepts and the locator point (h,k), then write the equations in graphing form.

  1. y = 7 + 2x2 + 4x– 5
  2. x2 = 2x + x(2x– 4) + y

2-129.Consider the system of equations at right:

  1. What is the parent of each equation?
  2. Solve this system algebraically.
  3. Find where the two graphs intersect.
  4. Explain the relationship between parts (b) and (c) above.

2-130.Write an equation for each of the following sequences.

  1. 10, 2.5, 0.625, …
  2. –2, –8, –14, …

2-131.Find the intercepts, the locator point (h, k),the domain, and the range for each of the following functions.

Lesson 2.2.3

  • 2-121.See below:
  • A vertical translation of a distance k.
  • A reflection across the x-axis.
  • A horizontal translation of a distance h.
  • A vertical dilation with a stretch (or shrink) factor of a. If a is negative, there is also a reflection across the x-axis.
  • 2-122.See below:
  • See sample graphs below.
  • f(–x) = (–x)2 = x2, f(–x) = (–x)3 =–x3,, for x ≤ 0 cannot be simplified,,
  • A reflection across the y-axis.
  • 2-123.See below:
  • x2 and
  • x3,
  • bx,·is niether because f(x) =has the domianx ≥ 0, and for that domain f(–x) is defined only for x = 0. Thus f(–x) neither equals f(x) nor–f(x).
  • 2-125. See below:
  • Neither
  • Neither
  • Even
  • 2-126.See below:
  • See graph below.
  • See graph below.
  • Neither function is odd nor even.
  • 2-127. y =–(x– 2)2 + 3
  • 2-128. See below:
  • x: (–1, 0), y: (0, 2), V: (–1, 0), y = 2(x + 1)2.
  • x: (0, 0), (2, 0), y: (0, 0), V: (1, 1) y =–(x– 1)2 + 1
  • 2-129.See below:
  • y = x
  • (,)
  • (,)
  • The solution to the system is the point at which the lines intersect.
  • 2-130.See below:
  • t(n)= 40()nor 10()n–1
  • t(n) =–6n + 4
  • 2-131.See below:
  • x: (2, 0), (6, 0) y: (0, 2), vertex: (4,–2), D: all real numbers, R: y ≥–2
  • x: (–4, 0), (2, 0), y: (0, 2), vertex (–1, 3), D: all real numbers, R: y ≤ 3