Q5 Investigate translations, reflections, stretches, and compressions of y = x²

Ontario Grade 10 Mathematics

Quadratic Relations

Translations, Reflections, Stretches, and Compressions of the Parabola

The parabola is one of the most useful curves in mathematics. Every time you toss a ball, the path it follows is roughly parabolic. In Grade 9 you worked with linear relations; now you will explore a non-linear relation — the quadratic — and discover how its graph changes shape and position when you adjust its equation. This lesson focuses on one parent curve,

y = x^2

, and shows exactly what happens when you translate (slide), reflect (flip), stretch (pull), or compress (squeeze) it. By the end you will be able to look at an equation and picture its graph, or look at a graph and write its equation.

What you will learn

Identify the key features of the base parabola y = x² and use them as a reference for every transformation.
Describe how adding or subtracting a constant shifts the parabola up, down, left, or right.
Explain how multiplying by a positive or negative value stretches, compresses, or reflects the parabola.
Combine multiple transformations to write or interpret an equation of the form y = a(x − h)² + k.
Sketch the transformed parabola and state its vertex, axis of symmetry, and direction of opening.

The Base Parabola y = x²

Before changing the graph, you need to know it well. The equation

y = x^2

produces a U-shaped curve called a parabola. It opens upward and is perfectly symmetric about the y-axis.

Three features describe every parabola. The vertex is the turning point — the lowest or highest point. For

y = x^2

the vertex is at the origin

(0, 0)

. The axis of symmetry is the vertical line that cuts the parabola in half; here it is the line

x = 0

. The direction of opening tells you whether the parabola opens up or down.

A useful reference table shows a few points on

y = x^2

: when

x = -2

,

y = 4

; when

x = -1

,

y = 1

; when

x = 0

,

y = 0

; when

x = 1

,

y = 1

; when

x = 2

,

y = 4

. Notice the symmetry: points to the left and right of the vertex always share the same y-value.

y = x^2

The vertex of y = x² is (0, 0).
The axis of symmetry is x = 0 (the y-axis).
The parabola opens upward; y is never negative.
The curve is symmetric: the left and right sides are mirror images.

Translations — Sliding the Parabola

A translation moves every point of the graph the same distance in the same direction without changing its shape. There are two kinds: vertical and horizontal.

Vertical translation: Adding a constant

k

to the equation shifts the parabola up when

k > 0

and down when

k < 0

. The equation becomes

y = x^2 + k

. The vertex moves from

(0, 0)

to

(0, k)

, and the axis of symmetry stays at

x = 0

. For example,

y = x^2 + 3

shifts every point up 3 units, so the vertex is at

(0, 3)

.

Horizontal translation: Replacing

x

with

(x - h)

shifts the parabola right when

h > 0

and left when

h < 0

. The equation becomes

y = (x - h)^2

. The vertex moves to

(h, 0)

. Be careful with the sign:

y = (x - 4)^2

shifts right 4 units (vertex at

(4, 0)

), while

y = (x + 4)^2

can be written as

y = (x - (-4))^2

, which shifts left 4 units (vertex at

(-4, 0)

).

When both translations are combined, the vertex lands at

(h, k)

and the equation becomes

y = (x - h)^2 + k

. This form is called vertex form and is very efficient for reading off the vertex directly.

y = (x - h)^2 + k

Adding k outside the bracket shifts the parabola vertically by k units.
Replacing x with (x − h) inside the bracket shifts the parabola horizontally by h units.
The sign inside the bracket is the opposite of the direction of the shift.
Vertex form y = (x − h)² + k places the vertex at (h, k).

Reflections — Flipping the Parabola

A reflection flips the graph over a line called the mirror line. For parabolas at this level, the relevant mirror line is the x-axis.

Multiplying the right side by

-1

changes

y = x^2

to

y = -x^2

. Every y-value that was positive becomes negative, so the parabola flips from opening upward to opening downward. The vertex stays at

(0, 0)

because the vertex y-value is zero, and multiplying zero by

-1

still gives zero.

In general, if

a < 0

in the equation

y = a x^2

, the parabola opens downward. If

a > 0

, it opens upward. The reflection is captured entirely by the sign of

a

.

y = -x^2

Multiplying by −1 reflects the parabola over the x-axis.
A negative value of a means the parabola opens downward.
The vertex is unaffected by a reflection when it sits on the mirror line.
The shape (width) stays the same under a pure reflection (when |a| = 1).

Stretches and Compressions — Changing the Width

Multiplying

x^2

by a positive number

a

(where

a \neq 1

) changes how wide or narrow the parabola looks. This is called a vertical stretch or vertical compression.

When

a > 1

, the parabola is vertically stretched: it becomes narrower, because the y-values grow faster. For example,

y = 3x^2

rises three times as quickly as

y = x^2

. When

0 < a < 1

, the parabola is vertically compressed: it becomes wider, because the y-values grow more slowly. For example,

y = \frac{1}{2}x^2

rises half as quickly as

y = x^2

.

A helpful way to see this: for the point where

x = 2

, the base parabola gives

y = 4

. With

a = 3

, it gives

y = 12

, which is much higher at the same x — so the curve looks narrower. With

a = \frac{1}{2}

, it gives

y = 2

, which is lower at the same x — so the curve looks wider.

The vertex and axis of symmetry do not move when only

a

changes (as long as translations are not also applied). The general equation combining all transformations is

y = a(x - h)^2 + k

, where

a

controls shape and direction,

h

controls horizontal position, and

k

controls vertical position.

y = a(x - h)^2 + k

When a > 1 the parabola is vertically stretched (narrower).
When 0 < a < 1 the parabola is vertically compressed (wider).
A negative a reflects and also stretches or compresses.
The vertex (h, k) is unchanged by changing a alone.
All four transformations combine in y = a(x − h)² + k.

Reading and Sketching the Full Transformation

Given any equation in the form

y = a(x - h)^2 + k

, you can read off four facts immediately: the vertex is

(h, k)

; the axis of symmetry is the vertical line

x = h

; the direction of opening is up if

a > 0

and down if

a < 0

; and the width compared to

y = x^2

is narrower if

|a| > 1

and wider if

|a| < 1

.

To sketch the graph, start by plotting the vertex. Then use the value of

a

to find two more points by substituting

x = h + 1

and

x = h - 1

into the equation. Reflect those points across the axis of symmetry and draw a smooth curve through all five points.

To write an equation from a described transformation, work backwards: identify the vertex to get

h

and

k

, determine the direction of opening for the sign of

a

, and use one known point to calculate the exact value of |a|.

Vertex is at (h, k); axis of symmetry is x = h.
Sign of a gives direction of opening.
|a| compared to 1 gives relative width.
Substitute x = h ± 1 to find two easy points for sketching.
Use a known point to solve for a when writing the equation.

Effect of Each Parameter in y = a(x − h)² + k

Parameter	What it controls	Effect when value increases	Effect when value decreases
a (positive)	Width and direction	Narrows (vertical stretch)	Widens (vertical compression toward 0)
a (negative)	Reflection + width	Reflected; narrows as \|a\| grows	Reflected; widens as \|a\| shrinks toward 0
h	Horizontal position	Vertex moves right	Vertex moves left
k	Vertical position	Vertex moves up	Vertex moves down

Worked example

Identifying All Transformations from an Equation

For the parabola y = −2(x − 3)² + 5, state the vertex, axis of symmetry, direction of opening, and whether the parabola is wider or narrower than y = x². Then find the y-coordinate when x = 4.

Match the equation to vertex form
The vertex form is $y = a(x - h)^2 + k$ . Compare $y = -2(x - 3)^2 + 5$ to this template. Reading off the values gives $a = -2$ , $h = 3$ , and $k = 5$ .
$a = -2,\ h = 3,\ k = 5$
State the vertex and axis of symmetry
The vertex is at $(h, k)$ , so the vertex is $(3, 5)$ . The axis of symmetry is the vertical line through the vertex, which is $x = 3$ .
$(h, k) = (3, 5)$
Determine direction of opening
Because $a = -2$ is negative, the parabola opens downward. This is a reflection of $y = x^2$ over the x-axis combined with the stretch.
$a = -2 < 0$
Compare width to y = x²
The width depends on |a|. Here $|a| = |-2| = 2$ . Since $2 > 1$ , the parabola is vertically stretched, meaning it appears narrower than $y = x^2$ .
$|a| = 2 > 1$
Find y when x = 4
Substitute $x = 4$ into the equation. First compute $(4 - 3)^2 = 1^2 = 1$ . Then multiply by $-2$ to get $-2$ . Finally add $5$ to get $y = 3$ .
$y = -2(4 - 3)^2 + 5 = -2(1) + 5 = 3$

Answer: Vertex:

(3, 5)

; axis of symmetry:

x = 3

; opens downward; narrower than

y = x^2

; when

x = 4

,

y = 3

.

Check: Verify the point

(4, 3)

: substitute back into

y = -2(x-3)^2 + 5

. We get

-2(1)^2 + 5 = -2 + 5 = 3

. Confirmed.

Worked example

Writing the Equation from a Description

A parabola opens upward, has its vertex at (−2, −4), and passes through the point (0, −2). Write its equation in the form y = a(x − h)² + k.

Set up vertex form using the vertex
The vertex is $(-2, -4)$ , so $h = -2$ and $k = -4$ . Substitute these into the vertex form template. Remember that $(x - h)$ becomes $(x - (-2)) = (x + 2)$ .
$y = a(x + 2)^2 - 4$
Substitute the known point to find a
The parabola passes through $(0, -2)$ , meaning when $x = 0$ , $y = -2$ . Substitute both values into the equation from Step 1.
$-2 = a(0 + 2)^2 - 4$
Simplify and solve for a
Compute $(0 + 2)^2 = 4$ . The equation becomes $-2 = 4a - 4$ . Add 4 to both sides to get $2 = 4a$ . Divide both sides by 4 to find $a = \frac{1}{2}$ .
$a = \frac{2}{4} = \frac{1}{2}$
Check the direction of opening
Since $a = \frac{1}{2} > 0$ , the parabola opens upward. This agrees with the given description, so the sign is correct.
$a = \frac{1}{2} > 0$
Write the final equation
Substitute $a = \frac{1}{2}$ , $h = -2$ , and $k = -4$ into the vertex form to get the final equation.
$y = \frac{1}{2}(x + 2)^2 - 4$

Answer:

y = \frac{1}{2}(x + 2)^2 - 4

Check: Check with the given point

(0, -2)

:

y = \frac{1}{2}(0 + 2)^2 - 4 = \frac{1}{2}(4) - 4 = 2 - 4 = -2

. The point satisfies the equation. Also

a = \frac{1}{2}

and

0 < \frac{1}{2} < 1

, so this parabola is wider than

y = x^2

and opens upward — consistent with the description.

Common mistakes and how to avoid them

Writing y = (x + 3)² and thinking the graph shifts right 3 units.

Correction: The horizontal shift is opposite to the sign inside the bracket. y = (x + 3)² = (x − (−3))², so h = −3 and the graph shifts left 3 units.

Thinking a larger value of a always makes the parabola wider.

Correction: A larger value of a (when a > 1) makes the parabola narrower, not wider, because y-values grow faster. Values of a between 0 and 1 make the parabola wider.

Moving the vertex when only a changes.

Correction: Changing a only stretches, compresses, or reflects the parabola. The vertex (h, k) does not move unless h or k also change.

Forgetting that a negative a reflects the parabola, so it now opens downward.

Correction: Always check the sign of a first. If a < 0, the parabola opens downward; the vertex is now the highest point, not the lowest.

Substituting the wrong x-value when checking a point — for example, using x = h instead of the given point.

Correction: To verify a point, substitute the x-coordinate of that specific point into the equation and confirm that the y-value you calculate matches the given y-coordinate.

Lesson summary

The base parabola y = x² has vertex (0, 0), axis of symmetry x = 0, and opens upward.
Adding k shifts the parabola vertically; replacing x with (x − h) shifts it horizontally. Together they move the vertex to (h, k).
Multiplying by a negative value reflects the parabola over the x-axis, causing it to open downward.
When |a| > 1 the parabola is narrower (vertical stretch); when 0 < |a| < 1 it is wider (vertical compression).
All four transformations combine in vertex form: y = a(x − h)² + k.
To find a unknown value of a, substitute the coordinates of any known point and solve algebraically.

Check your understanding

Question 1

What is the vertex of the parabola y = 3(x + 1)² − 7?

(1, −7)
(−1, 7)
(−1, −7)
(1, 7)

Show answer and explanation

(−1, −7)

Rewrite as y = 3(x − (−1))² + (−7). So h = −1 and k = −7, giving vertex (−1, −7). The sign inside the bracket is opposite to the shift direction.

Question 2

How does the graph of y = ¼x² compare to the graph of y = x²?

Narrower and opens downward
Wider and opens upward
Narrower and opens upward
Same width but shifted up

Show answer and explanation

Wider and opens upward

Since a = ¼ and 0 < ¼ < 1, the parabola is vertically compressed, making it wider. Since a > 0, it still opens upward. There is no translation, so there is no shift.

Question 3

A parabola has vertex (2, −3) and passes through (3, −1). Which equation matches?

y = 2(x − 2)² − 3
y = (x − 2)² − 3
y = −2(x − 2)² − 3
y = ½(x − 2)² − 3

Show answer and explanation

y = 2(x − 2)² − 3

Substitute (3, −1): y = a(3−2)² − 3 → −1 = a(1) − 3 → a = 2. The equation is y = 2(x − 2)² − 3. Check: 2(1)² − 3 = −1. Correct.

Question 4

Which transformation maps y = x² onto y = −x² + 6?

A horizontal shift of 6 units and a vertical stretch
A reflection over the x-axis and a vertical shift up 6 units
A reflection over the y-axis and a shift right 6 units
A vertical compression and a shift down 6 units

Show answer and explanation

A reflection over the x-axis and a vertical shift up 6 units

Multiplying by −1 reflects the parabola over the x-axis. Adding 6 shifts it up 6 units. The y-axis is the axis of symmetry for both y = x² and y = −x², so no horizontal shift or compression is involved.

Key terms

Parabola: The U-shaped (or ∩-shaped) curve produced by a quadratic relation such as y = x².
Vertex: The turning point of a parabola — its lowest point when it opens upward, or its highest point when it opens downward.
Axis of symmetry: The vertical line that divides the parabola into two mirror-image halves. Its equation is x = h.
Vertex form: The equation y = a(x − h)² + k, which directly shows the vertex (h, k) and the value of a.
Translation: A transformation that slides every point of a graph the same distance in the same direction, changing position but not shape.
Reflection: A transformation that flips a graph over a mirror line, reversing the direction of opening for a parabola.
Vertical stretch: A transformation where |a| > 1 makes the parabola narrower by multiplying every y-value by a factor greater than 1.
Vertical compression: A transformation where 0 < |a| < 1 makes the parabola wider by multiplying every y-value by a factor less than 1.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic Q5. It is a study resource, not an official curriculum publication.

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Q5 · Investigate translations, reflections, stretches, and compressions of y = x²

Ontario Grade 10 Mathematics

Translations, Reflections, Stretches, and Compressions of the Parabola

What you will learn

The Base Parabola y = x²

Translations — Sliding the Parabola

Reflections — Flipping the Parabola

Stretches and Compressions — Changing the Width

Reading and Sketching the Full Transformation

Effect of Each Parameter in y = a(x − h)² + k

Identifying All Transformations from an Equation

Writing the Equation from a Description

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

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About this lesson