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Quadratic Relations of the Form y = ax² + bx + c · Quadratic Relations and Their Graphs

Changing the Shape: Stretches, Compressions and Reflections of y = ax²

2.1About 65 minutes
Learning goals
  • I can describe the effect of a on the graph of y = ax²: vertical stretch, vertical compression and reflection in the x-axis.
  • I can map points from y = x² to y = ax².
  • I can use the step pattern to plot y = ax² quickly.
  • I can find a when I know a point on a parabola with vertex (0, 0).

Words to know

vertical stretch
A transformation that multiplies every y-value by a number greater than 1, making a parabola narrower.
vertical compression
A transformation that multiplies every y-value by a number between 0 and 1, making a parabola wider.
reflection in the x-axis
A flip over the x-axis: every point (x, y) moves to (x, −y).
factor
The number every y-value is multiplied by in a stretch or compression; for y = ax² it is a (or the size of a when a is negative).
step pattern
The rises 1, 3, 5, … of y = x² from its vertex for each step of 1 to the right, multiplied by a for y = ax².

Changing the shape

Translations move a parabola without changing its shape. The number a in y = ax² does something different: it reshapes the parabola. It can make it narrower, wider, or flip it upside down. In this lesson you will investigate what a does, and learn a quick step pattern for plotting any parabola.

Investigate with technology

In Desmos type y = ax^2 with a slider for a from −5 to 5. Drag a slowly and watch the point at x = 2.

  • For y = x², the point is (2, 4). For y = 2x², it is (2, 8): every y-value is doubled. The parabola is pulled away from the x-axis and looks narrower. This is a vertical stretch by a factor of 2.
  • For y = 0.5x², the point is (2, 2): every y-value is halved. The parabola looks wider. This is a vertical compression by a factor of 0.5.
  • For y = −x², the point is (2, −4): every y-value changes sign. The parabola is reflected in the x-axis and opens down.

Every point (x, y) on y = x² maps to (x, ay). The vertex (0, 0) does not move, because a × 0 = 0.

xy−4−3−2−101234−10123456789(2, 8)(2, 4)(2, 2)y = 2x²y = x²y = 0.5x²
y = 2x² is a vertical stretch of y = x² (dashed) by a factor of 2: (2, 4) moves to (2, 8). y = 0.5x² is a vertical compression by a factor of 0.5: (2, 4) moves to (2, 2).
xy−3−2−10123−9−6−30369(2, 4)(2, −4)y = x²y = −x²y = −3x²
A negative a reflects the parabola in the x-axis: (2, 4) on y = x² becomes (2, −4) on y = −x². y = −3x² is reflected and stretched by a factor of 3.

Summary of a

  • a > 1: vertical stretch by a factor of a (narrower), opens up.
  • 0 < a < 1: vertical compression by a factor of a (wider), opens up.
  • a < 0: reflection in the x-axis (opens down), plus a stretch if a < −1 or a compression if −1 < a < 0.
  • a = 1: no change. a = −1: reflection only.

The vertex (0, 0) and the axis of symmetry x = 0 stay the same.

Stretch is vertical

A "vertical stretch" makes the parabola look narrower, which surprises some students. Think of grabbing the parabola's arms and pulling them up: the sides get steeper, so the U closes in.

The step pattern

Starting at the vertex of y = x² and moving 1 unit right each time, the graph goes up 1, then 3, then 5 (the first differences: see the left panel below). For y = ax², multiply each step by a:

  • y = x²: over 1 up 1, over 1 up 3, over 1 up 5
  • y = 2x²: over 1 up 2, over 1 up 6, over 1 up 10
  • y = −0.5x²: over 1 down 0.5, over 1 down 1.5, over 1 down 2.5

Do the same steps to the left, because the parabola is symmetric.

xy01230246810135y = x²: over 1, up 1, 3, 5xy01230246810121416182610y = 2x²: over 1, up 2, 6, 10
The step pattern from the vertex. For y = x² the rises are 1, 3, 5. For y = 2x² every rise is doubled: 2, 6, 10.
Worked example

Example 1: Describe and plot y = 3x²

Describe the transformation and use the step pattern to list seven points on y = 3x².

  1. a = 3 > 1: a vertical stretch by a factor of 3. It opens up. Vertex (0, 0).
  2. Steps from the vertex: up 3 × 1 = 3, 3 × 3 = 9, 3 × 5 = 15.
  3. Right side: (1, 3), (2, 3 + 9) = (2, 12), (3, 12 + 15) = (3, 27).
  4. Left side by symmetry: (−1, 3), (−2, 12), (−3, 27).
  5. Check with the mapping (x, y) → (x, 3y): (2, 4) → (2, 12). ✓

Vertical stretch by 3; points (−3, 27), (−2, 12), (−1, 3), (0, 0), (1, 3), (2, 12), (3, 27).

Worked example

Example 2: Describe y = −0.25x²

Describe the transformations, and find the points with x = ±2 and x = ±4.

  1. a = −0.25 is negative, so there is a reflection in the x-axis (it opens down).
  2. The size of a is 0.25 < 1, so there is also a vertical compression by a factor of 0.25 (it is wider than y = −x²).
  3. x = ±2: −0.25(4) = −1, giving (2, −1) and (−2, −1).
  4. x = ±4: −0.25(16) = −4, giving (4, −4) and (−4, −4).

Reflection in the x-axis and vertical compression by 0.25; points (±2, −1) and (±4, −4).

Worked example

Example 3: Finding a from a point

A parabola with vertex (0, 0) passes through (2, −12). Find its equation.

  1. The form is y = ax² (vertex at the origin).
  2. Substitute the point: −12 = a(2)², so −12 = 4a.
  3. a = −3.

y = −3x² (reflected in the x-axis and stretched by a factor of 3).

Just for fun

A funny-but-true fact: a parabola-shaped mirror sends all the light that comes straight into it to one point, called the focus. That is why many satellite dishes and some solar cookers are parabola-shaped. A parabolic dish that gathers sunlight onto one spot can get hot enough to cook food.

Hands-on activity

Stretch it on grid paper

You need: graph paper, a pencil, and your parabola template from earlier (optional).

  1. Draw axes with x from −3 to 3 and y from −18 to 18 (use 1 square for 2 units on the y-axis).
  2. Plot y = x² using the step pattern 1, 3, 5. Plot y = 2x² using 2, 6, 10. Plot y = 0.5x² using 0.5, 1.5, 2.5. Plot y = −x² using −1, −3, −5.
  3. Try to lay your y = x² template (if you have one) on top of y = 2x². Does it fit?
  4. Complete: "Changing a changes the ______ of the parabola, but not its ______."

What to look for: the template fits y = −x² if you flip it over, but it does not fit y = 2x² or y = 0.5x², because a stretch or compression changes the shape. A good completed sentence: "Changing a changes the shape (width) and direction of the parabola, but not its vertex or axis of symmetry."

Your turn

1

Describe the transformation from y = x² to y = 4x².

2

Describe the transformation from y = x² to y = 0.2x².

3

Describe the transformation from y = x² to y = −2x².

4

Use the step pattern to find the points of y = 0.5x² at x = 1, 2 and 3.

5

The point (3, 9) is on y = x². Where does it move under y = −4x²?

6

A parabola with vertex (0, 0) passes through (−1, 5). Find a.

More practice: warm-up

1

Is y = 6x² narrower or wider than y = x²?

2

Does y = −0.7x² open up or down?

3

What is the step pattern for y = x²?

4

Write the mapping rule for y = 3x².

5

Does changing a move the vertex of y = ax²?

More practice: on your own

1

List the points at x = 0, ±1, ±2 for y = −1.5x².

2

Order from widest to narrowest: y = 3x², y = 0.5x², y = −x², y = −5x².

3

A parabola with vertex (0, 0) passes through (4, 8). Find its equation.

4

Find the second differences of y = 3x² (x-steps of 1) and connect them to a.

5

A student says y = −2x² is a compression because the graph goes down. Correct them.

Challenge

1

A satellite dish cross-section is modelled by y = ax² with x and y in centimetres. The dish is 80 cm wide and 10 cm deep. Find a.

2

Are y = 2x² and y = −2x² congruent? Explain.

3

For which values of a is y = ax² wider than y = x² and opening down?

Check your understanding

Quiz

1.

Compared with y = x², the graph of y = 5x² is…

2.

Which describes y = −0.5x²?

3.

Under y = 3x², the point (−2, 4) on y = x² moves to…

4.

The step pattern for y = 2x² from the vertex is…

5.

A parabola with vertex (0, 0) passes through (3, −18). Its equation is…

6.

Which transformation keeps the vertex of y = x² at (0, 0)?

For parents and teachers

Big idea: Expectation 2.1 also covers a: in y = ax², a stretches (|a| > 1) or compresses (0 < |a| < 1) the parabola vertically and, if negative, reflects it in the x-axis. The step pattern (1, 3, 5 times a) is the main sketching tool for the next lessons.

How to help: Ask your teen to predict, before graphing, whether a parabola will be narrower or wider and which way it opens. Have them say the step pattern out loud for y = 2x² (2, 6, 10) and y = 0.5x² (0.5, 1.5, 2.5).

Common difficulties:

  • Calling a stretch "wider". A vertical stretch makes the U narrower.
  • Thinking a negative a means compression. The sign controls reflection; the size controls stretch or compression.
  • Applying a to x instead of y (writing (2a, 4) instead of (2, 4a)).