Transformations: shifts and reflections

About this lesson

Integrated Math I › Functions and sequences

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Transformations: shifts and reflections

Warm-up

Three quick questions from earlier lessons.

Discover

Move the sliders and watch the graph of y=(x−h)2+ky = (x - h)^2 + k.

The graph of y = (x − h)² + k, a parabola whose vertex is at (h, k) and moves with the sliders.

When hh goes from 0 to 2, which way does the graph move?

  1. To the right
  2. To the left

Key idea

Changing the rule moves or flips the graph without changing its shape.

y=f(x−h)+ky = f(x - h) + k moves the graph of ff by hh units to the right and kk units up. y=−f(x)y = -f(x) reflects it across the xx-axis; y=f(−x)y = f(-x) reflects it across the yy-axis.

A change inside f(… )f(\dots) acts on xx (horizontally); a change outside acts on yy (vertically).

Worked example

Describe y=(x+1)2−3y = (x + 1)^2 - 3 compared with y=x2y = x^2.

1 unit to the left
x + 1 = x − (−1): h = −1.
3 units down
− 3 outside: k = −3.
Vertex: (−1,−3)(-1, -3)
The vertex (0, 0) moves with the graph.

Your turn, guided

Your turn: y=(x−4)2+2y = (x - 4)^2 + 2.

4 units to the right
x − 4: h = 4.
2 units up
+ 2 outside.
The vertex is at x=4x = 4
The vertex moves from x = 0 to x = …

Practice ★

  1. The graph of y=x2y = x^2 is moved 1 units to the right and 6 units down. Write the equation of the new graph.
  2. The graph of y=x2y = x^2 is moved 3 units to the right and 1 units down. Write the equation of the new graph.
  3. The graph of y=x2y = x^2 is moved 4 units to the right and 6 units down. Write the equation of the new graph.
  4. The graph of y=x2y = x^2 is moved 7 units to the right and 1 units down. Write the equation of the new graph.

Watch out!

Practice ★★

  1. Draw the graph of y=(x−1)2−3y = (x - 1)^2 - 3: place the vertex and one more point.
  2. Draw the graph of y=(x−1)2−1y = (x - 1)^2 - 1: place the vertex and one more point.
  1. Match each equation with what it does to the graph of y=f(x)y = f(x).
    1. y=f(x)+3y = f(x) + 3
    2. y=f(x−3)y = f(x - 3)
    3. y=−f(x)y = -f(x)
    4. y=f(−x)y = f(-x)
    1. moves it up 3 units
    2. moves it 3 units to the right
    3. reflects it across the x-axis
    4. reflects it across the y-axis

Exit ticket

Three questions to check you have got it.

  1. The graph of y=x2y = x^2 is moved 1 units to the right and 6 units down. Write the equation of the new graph.
  2. The graph of y=x2y = x^2 is moved 3 units to the right and 1 units down. Write the equation of the new graph.
  3. The graph of y=x2y = x^2 is moved 4 units to the right and 6 units down. Write the equation of the new graph.

Summary card

Shifts and reflections

  • f(x−h)f(x - h): hh units right. f(x)+kf(x) + k: kk units up.
  • −f(x)-f(x): flip across the xx-axis. f(−x)f(-x): flip across the yy-axis.
  • Inside f(… )f(\dots) acts on xx; outside acts on yy.

Watch out: f(x+3)f(x + 3) moves the graph left.

End of the lesson

Next lesson: Sequences: explicit and recursive formulas Back to the chapter

Answers: Transformations: shifts and reflections

A

  1. (x−1)2−6\left(x - 1\right)^{2} - 6
  2. (x−3)2−1\left(x - 3\right)^{2} - 1
  3. (x−4)2−6\left(x - 4\right)^{2} - 6
  4. (x−7)2−1\left(x - 7\right)^{2} - 1

B

  1. y=(x−1)2−3y = \left(x - 1\right)^{2} - 3
  2. y=(x−1)2−1y = \left(x - 1\right)^{2} - 1

C

  1. y=f(x)+3y = f(x) + 3 → moves it up 3 units · y=f(x−3)y = f(x - 3) → moves it 3 units to the right · y=−f(x)y = -f(x) → reflects it across the x-axis · y=f(−x)y = f(-x) → reflects it across the y-axis

D

  1. (x−1)2−6\left(x - 1\right)^{2} - 6
  2. (x−3)2−1\left(x - 3\right)^{2} - 1
  3. (x−4)2−6\left(x - 4\right)^{2} - 6

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