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AI HL · 12.5

Functions

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12.5.6

Stretches and reflections

12.5.6

Stretches and reflections

Target: I can sketch y = p f(x) and say it is a vertical stretch with scale factor p, including p less than 1. (+4 more)

By the end of this section

Key points

Outside the bracket changes y

  • \(y=p\,f(x)\) is a vertical stretch, scale factor \(p\). Every y-coordinate is multiplied by \(p\). \(p\) can be a fraction less than 1.
  • \(y=-f(x)\) is a reflection in the x-axis. Every y-coordinate is multiplied by −1.
  • Points on the x-axis do not move in a vertical stretch.

Inside the bracket changes x

  • \(y=f(qx)\) is a horizontal stretch, scale factor \(\frac1q\). Every x-coordinate is divided by \(q\).
  • \(y=f(-x)\) is a reflection in the y-axis. Every x-coordinate is multiplied by −1.
  • Warning: \(f(2x)\) is half as wide, not twice as wide. The scale factor is \(\frac12\).
  • Points on the y-axis do not move in a horizontal stretch.

Describing a combination

  • Read \(p\,f(qx)\) in two halves: \(p\) for the vertical stretch (a minus sign is a reflection in the x-axis), \(q\) for the horizontal one (a minus sign is a reflection in the y-axis).
  • A point \((a, b)\) on f moves to \(\left(\frac aq,\ pb\right)\).
  • To compare two quadratics, write the new equation as \(p\,f(x)\) or \(f(qx)\) and read off the scale factor.

On the calculator

Checking a transformation on a graph

  • Enter f1(x) as the original function.
  • Enter f2(x) = 2·f1(x), then f3(x) = f1(2x), and compare the graphs with your sketch.
  • Trace to a key point and check its new coordinates.

Screenshot and video to come

Explore: transformations of a graph

The dashed graph is y = f(x). The orange graph is y = p f(qx + c) + b. Drag the first orange point to translate the graph, or choose a kind of change and move the sliders. Say what happens to the labelled points before you read the answer.

Move a slider to change the graph.

  1. Choose Translation and drag A′ 3 units to the left. What is c now? Why is that the opposite of what the plus sign suggests?
  2. Choose Stretch and set q = 2. Is the graph twice as wide or half as wide? Then choose Reflection and tick the y-axis box. Which coordinate of each labelled point changes sign?
  3. Choose All and set c = 2 and q = 2. Read the order in the sentence. Does the graph move 2 units, or a different amount? Why?
  4. Choose 1/x or eˣ and translate the graph. Where does each asymptote go? Which numbers move the vertical one, and which the horizontal one?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A vertical stretch

The graph of \(y=f(x)\) is shown. It has its minimum at (2, −1), crosses the x-axis at x = 1 and x = 3, and crosses the y-axis at (0, 3). Sketch \(y=2f(x)\) and give its y-intercept.

−2−112345−3−2−112345xyy = f(x)
Working
  1. Outside the bracket

    \(y=2f(x)\)

    The 2 multiplies the outputs, so only the y-coordinates change. Vertical stretch, scale factor 2.

  2. −2−112345−4−22468xyy = f(x)y = 2f(x)
    Move the key points

    Minimum (2, −1) → (2, −2); x-intercepts 1 and 3 stay; y-intercept 3 → 6

    Points on the x-axis have y = 0, so they do not move.

  3. Sketch

    y-intercept: (0, 6)

    A fraction such as ⅓ gives a squashed graph. The rule is the same.

Video walkthrough coming
A horizontal stretch

The graph of \(y=f(x)\) is shown. It has its minimum at (1, −4), crosses the x-axis at x = −1 and x = 3, and crosses the y-axis at (0, −3). Sketch \(y=f(\frac{x}{2})\) and find where it meets the x-axis.

−3−2−112345−6−5−4−3−2−112xyy = f(x)
Working
  1. Inside the bracket

    \(y=f(qx)\) with \(q=\frac12\)

    Inside the bracket changes x. The scale factor is \(\frac1q=2\).

  2. −4−22468−6−5−4−3−2−112xyy = f(x)y = f(x/2)
    Move the key points

    Minimum (1, −4) → (2, −4); x-intercepts −1 → −2, 3 → 6

    Multiply every x-coordinate by 2. The y-coordinates stay.

  3. The y-intercept

    (0, −3) stays

    x = 0 does not move when you multiply it by 2.

Video walkthrough coming
Which stretch?

Describe the transformation that maps \(y=2x^2-6x+1\) onto \(y=5x^2-15x+2.5\).

Working
  1. Name the first graph

    \(f(x)=2x^2-6x+1\)

    Call the starting graph f.

  2. Write the target with f

    \(5x^2-15x+2.5=2.5(2x^2-6x+1)=2.5f(x)\)

    Every term of f is multiplied by the same number.

  3. State it

    Vertical stretch, scale factor 2.5

    The number is outside the bracket, so it is vertical.

Video walkthrough coming
Reflect in the x-axis

The graph of \(y=f(x)\) has a vertex at (−4, 3). Find the vertex of \(y=-f(x)\).

Working
  1. Outside the bracket

    \(y=-f(x)\)

    The minus is outside, so it acts on the outputs: a reflection in the x-axis.

  2. Multiply y by −1

    (−4, 3) → (−4, −3)

    A maximum becomes a minimum.

Video walkthrough coming
Reflect in the y-axis

The graph of \(y=f(x)\) has a vertex at (7, −3). Find the vertex of \(y=f(-x)\).

Working
  1. Inside the bracket

    \(y=f(-x)\)

    The minus is inside, so it acts on x: a reflection in the y-axis.

  2. Multiply x by −1

    (7, −3) → (−7, −3)

    The height does not change.

Video walkthrough coming
A combination

The point (4, −5) lies on the graph of \(y=f(x)\). Describe \(y=4f(-\frac{x}{2})\) as stretches and reflections, then find the image of (4, −5).

Working
  1. Outside the bracket

    4: vertical stretch, scale factor 4

    The 4 multiplies the outputs.

  2. Inside the bracket

    \(-\frac{x}{2}\): reflection in the y-axis and horizontal stretch, scale factor 2

    The minus reflects. The ½ gives scale factor 1/q = 2.

  3. Move the point

    x: 4 ÷ (−½) = −8; y: 4 × (−5) = −20

    Divide x by q, multiply y by p.

  4. Image

    (4, −5) → (−8, −20)

    Check the signs: x flips, y keeps its sign because p is positive.

Video walkthrough coming