Unit 1 · Topic 1.12

Transformations of Functions

g(x) = af(b(x − h)) + k dilates and reflects f by a and b and translates it h right and k up.

Practice 1: Procedural and Symbolic FluencyPractice 2: Multiple Representations

Part 1 · Hook

Why this matters

A designer on a phone app drags a curve 3 pixels right, stretches it to twice its height, then flips it. The app never redraws the curve from scratch; it changes the formula. Each drag or stretch is one small edit to the function.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. If f(x)=x2f(x) = x^2, what is f(x−3)f(x - 3)?

  1. (x−3)2(x - 3)^2
  2. x2−3x^2 - 3
  3. x2−9x^2 - 9
  4. x2+3x^2 + 3
Show the answer

Replace every xx with x−3x - 3: f(x−3)=(x−3)2f(x - 3) = (x - 3)^2.

  • Correct: (x−3)2(x - 3)^2:
  • x2−3x^2 - 3:
  • x2−9x^2 - 9:
  • x2+3x^2 + 3:

2. What is the vertex of y=(x+1)2−4y = (x + 1)^2 - 4?

  1. (−1,−4)(-1, -4)
  2. (1,−4)(1, -4)
  3. (−1,4)(-1, 4)
  4. (1,4)(1, 4)
Show the answer

Vertex form y=(x−h)2+ky = (x - h)^2 + k has vertex (h,k)(h, k); here h=−1h = -1 and k=−4k = -4.

  • Correct: (−1,−4)(-1, -4):
  • (1,−4)(1, -4):
  • (−1,4)(-1, 4):
  • (1,4)(1, 4):

Part 4 · See it

See it first

The parent graph f(x) = x^2 with vertex (0, 0) and the transformed graph g(x) = 2f(x - 3) - 1, narrower, with vertex (3, -1); the point (1, 1) on f moves to (4, 1) on g.
The parent f(x)=x2f(x) = x^2 and g(x)=2f(x−3)−1g(x) = 2f(x - 3) - 1. Each point (a,b)(a, b) on ff moves to (a+3,2b−1)(a + 3, 2b - 1) on gg. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Add a constant outside: f(x)+kf(x) + kevery output moves by kk (a vertical translation)
  2. Add a constant inside: f(x+h)f(x + h)the graph moves hh units left when h>0h > 0 (a horizontal translation)
  3. Multiply outside: af(x)a f(x)outputs are multiplied by aa (a vertical dilation; reflection across the xx-axis if a<0a < 0)
  4. Multiply inside: f(bx)f(bx)inputs are scaled by 1b\frac{1}{b} (a horizontal dilation; reflection across the yy-axis if b<0b < 0)

Part 6 · Key ideas

Key ideas

  • Changes outside the function act on outputs and move the graph up, down, or stretch it vertically. Changes inside act on inputs and work in reverse.
  • f(x−3)f(x - 3) moves the graph 33 right: the old output at 00 now appears at x=3x = 3.
  • In af(x)+ka f(x) + k, dilate first, then shift. Domain changes come from inside changes; range changes come from outside changes.

Part 7 · Misconception

A common mistake

The wrong idea: f(x+4)f(x + 4) moves the graph 44 units right, because of the plus sign.

What actually happens: It moves 44 left: g(−4)=f(0)g(-4) = f(0), so the point that was at input 00 is now at input −4-4.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

A parent graph ff and two transformations, gg and hh

−6−5−4−3−2−11234−4−3−2−1123456xyfgh

1. The graph shows the parent f(x)=x2f(x) = x^2 and a transformation gg with vertex (−2,−3)(-2, -3). Which equation gives gg?

  1. g(x)=f(x−2)−3g(x) = f(x - 2) - 3
  2. g(x)=f(x+2)−3g(x) = f(x + 2) - 3
  3. g(x)=f(x+2)+3g(x) = f(x + 2) + 3
  4. g(x)=f(x−3)−2g(x) = f(x - 3) - 2
Show the answer

The vertex moves from (0,0)(0, 0) to (−2,−3)(-2, -3): 22 left and 33 down. A shift left by 22 replaces xx with x+2x + 2, and down 33 subtracts 33 from the output: g(x)=f(x+2)−3g(x) = f(x + 2) - 3.

  • g(x)=f(x−2)−3g(x) = f(x - 2) - 3: Inside the parentheses the sign is opposite the direction: x−2x - 2 moves the graph right.
  • Correct: g(x)=f(x+2)−3g(x) = f(x + 2) - 3: Left 22 is x+2x + 2 inside; down 33 is −3-3 outside.
  • g(x)=f(x+2)+3g(x) = f(x + 2) + 3: Adding 33 outside moves the graph up, but the vertex is below the xx-axis.
  • g(x)=f(x−3)−2g(x) = f(x - 3) - 2: This mixes up the two shifts: the horizontal one goes inside, the vertical one outside.

2. The graph of hh is the graph of f(x)=x2f(x) = x^2 reflected across the xx-axis, vertically dilated by a factor of 12\frac{1}{2}, and then shifted up 44. Write h(x)h(x) in terms of xx.

Type an expression for h(x)=h(x) =.

Show the answer

Reflect and dilate: −12f(x)=−12x2-\frac{1}{2}f(x) = -\frac{1}{2}x^2. Then shift up 44: h(x)=−12x2+4h(x) = -\frac{1}{2}x^2 + 4. Check with the graph: h(2)=−2+4=2h(2) = -2 + 4 = 2.

  • Answer: h(x)=h(x) = −x22+4\frac{-x^{2}}{2} + 4

3. The point (1,1)(1, 1) is on the graph of ff. Which point is on the graph of g(x)=f(x+2)−3g(x) = f(x + 2) - 3?

  1. (3,−2)(3, -2)
  2. (−1,4)(-1, 4)
  3. (3,4)(3, 4)
  4. (−1,−2)(-1, -2)
Show the answer

If f(1)=1f(1) = 1, then g(−1)g(-1)=f(−1+2)−3= f(-1 + 2) - 3=f(1)−3= f(1) - 3=−2= -2. Every input moves 22 left and every output moves 33 down: (1,1)→(−1,−2)(1, 1) \to (-1, -2).

  • (3,−2)(3, -2): This moves the input 22 right; x+2x + 2 inside moves the graph left.
  • (−1,4)(-1, 4): This adds 33 to the output; −3-3 outside moves it down.
  • (3,4)(3, 4): Both shifts go the wrong way.
  • Correct: (−1,−2)(-1, -2): Input 1−2=−11 - 2 = -1, output 1−3=−21 - 3 = -2.

4. How does the graph of y=f(2x)y = f(2x) compare with the graph of y=f(x)y = f(x)?

  1. It is stretched horizontally by a factor of 22
  2. It is a horizontal dilation by a factor of 12\frac{1}{2}
  3. It is stretched vertically by a factor of 22
  4. It is shifted left by 22
Show the answer

f(2x)f(2x) reaches each output at half the input: the point (a,b)(a, b) on ff moves to (a2,b)(\frac{a}{2}, b). That is a horizontal dilation by a factor of 12\frac{1}{2}.

  • It is stretched horizontally by a factor of 22: The factor inside acts on inputs in reverse: 2x2x halves them.
  • Correct: It is a horizontal dilation by a factor of 12\frac{1}{2}: Each input is multiplied by 12\frac{1}{2}: the graph is compressed toward the yy-axis.
  • It is stretched vertically by a factor of 22: A vertical dilation multiplies outputs: that would be 2f(x)2f(x).
  • It is shifted left by 22: A shift adds to the input, as in f(x+2)f(x + 2).

5. The graph of ff has domain [0,4][0, 4] and range [−1,3][-1, 3]. Let g(x)=−2f(x−1)g(x) = -2f(x - 1). What are the domain and range of gg?

  1. Domain [1,5][1, 5], range [−6,2][-6, 2]
  2. Domain [−1,3][-1, 3], range [−6,2][-6, 2]
  3. Domain [1,5][1, 5], range [−2,6][-2, 6]
  4. Domain [0,4][0, 4], range [−6,2][-6, 2]
Show the answer

Domain: x−1x - 1 must be in [0,4][0, 4], so xx is in [1,5][1, 5]. Range: multiply the outputs −1-1 and 33 by −2-2 to get 22 and −6-6, so the range is [−6,2][-6, 2].

  • Correct: Domain [1,5][1, 5], range [−6,2][-6, 2]: Shift the inputs right 11; multiply the outputs by −2-2 and reorder.
  • Domain [−1,3][-1, 3], range [−6,2][-6, 2]: This shifts the domain left; x−1x - 1 moves it right.
  • Domain [1,5][1, 5], range [−2,6][-2, 6]: This dilates by 22 but forgets the reflection.
  • Domain [0,4][0, 4], range [−6,2][-6, 2]: A horizontal shift changes the domain.

6. Let f(x)=x2f(x) = x^2, with range [0,∞)[0, \infty). Which transformations of ff have a different range from ff? Select all that apply.

  1. y=f(x−5)y = f(x - 5)
  2. y=f(x)+5y = f(x) + 5
  3. y=−f(x)y = -f(x)
  4. y=f(3x)y = f(3x)
Show the answer

Vertical changes (shifts, dilations, reflections across the xx-axis) act on outputs, so they can change the range. Horizontal changes act on inputs and keep the range here.

  • y=f(x−5)y = f(x - 5): A shift right keeps the outputs [0,∞)[0, \infty).
  • Correct: y=f(x)+5y = f(x) + 5: Adding 55 moves the range to [5,∞)[5, \infty).
  • Correct: y=−f(x)y = -f(x): Negating gives (−∞,0](-\infty, 0].
  • y=f(3x)y = f(3x): A horizontal dilation keeps the outputs [0,∞)[0, \infty).

Part 9 · Summary

Summary

g(x)=af(b(x−h))+kg(x) = a f(b(x - h)) + k dilates and reflects ff by aa and bb and translates it hh right and kk up. Outside changes act on outputs and the range; inside changes act on inputs and the domain, in reverse.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic