Dilations of Quadratic Functions

Q: Why does a bigger $a$ make the parabola look narrower?

Because the outputs climb faster. On $y = x^{2}$, moving from $x = 1$ to $x = 2$ takes $y$ from $1$ to $4$. On $y = 2x^{2}$ it goes from $2$ to $8$. The graph shoots up sooner, which reads as a skinnier U.

Q: How is a dilation different from a translation?

A translation slides the graph to a new spot without changing its shape. A dilation keeps the vertex of the parent function in place but stretches or compresses the shape itself.

Q: What does a negative $a$ do?

It reflects the parabola across the $x$-axis, so it opens downward. The absolute value of $a$ still controls width: $|a| > 1$ is narrower and $|a| < 1$ is wider, reflected or not.

Q: Why do horizontal dilations work backwards?

The multiplier changes the input before squaring. In $f(2x)$, plugging in $x = 1$ computes $f(2)$ — the function reaches its values twice as fast, squeezing the graph toward the $y$-axis even though $2 > 1$.

A translation slides a parabola; a dilation reshapes it. Starting from the parent function $f(x) = x^{2}$ , multiplying by a constant stretches the graph, compresses it, or flips it over — and which one happens depends entirely on the size and sign of that constant.

There are two places the multiplier can live. Multiplying the whole function, $g(x) = a \cdot f(x)$ , is a vertical dilation. Multiplying the input, $g(x) = f(bx)$ , is a horizontal dilation. Learn to read both and you can describe any parabola's shape at a glance.

Vertical stretch and compression

In $g(x) = a \cdot f(x)$ , every output gets multiplied by $a$ . If $a > 1$ , outputs grow faster, so the parabola is vertically stretched and looks narrower. If $0 < a < 1$ , outputs shrink, so the parabola is vertically compressed and looks wider.

In the graph below, the black curve is $y = x^{2}$ , the blue curve is $y = 2x^{2}$ (stretched, narrower), and the red curve is $y = \dfrac{1}{2}x^{2}$ (compressed, wider). All three share the same vertex — a dilation from the parent does not move the vertex at the origin.

Horizontal dilations

In $g(x) = f(bx)$ , the multiplier sits on the input. It works in the reverse direction of what you might guess: if $b > 1$ , the graph is horizontally compressed and looks narrower, and if $0 < b < 1$ , the graph is horizontally stretched and looks wider.

The reason is that $b$ speeds up or slows down the input. In $f(2x)$ , the function reaches each output twice as fast, so the graph gets squeezed toward the $y$ -axis.

Reflections

When $a$ is negative in $g(x) = a \cdot f(x)$ , the graph also reflects across the $x$ -axis — the parabola flips to open downward. The size of $a$ still controls the shape: $y = -2x^{2}$ is reflected and narrower, while $y = -\dfrac{1}{2}x^{2}$ is reflected and wider.

In the graph below, the black curve is $y = x^{2}$ and the blue curve is $y = -2x^{2}$ : the negative sign flips the parabola downward, and $|a| = 2$ makes the reflected curve narrower.

Worked examples

Example 1: a vertical stretch

Describe $g(x) = 2f(x)$ compared to $f(x) = x^{2}$ .

The multiplier is on the whole function — vertical dilation

a = 2

Compare to

1

|2| > 1

Bigger than

1

means a vertical stretch — the graph is narrower

Answer: Vertical stretch; the parabola is narrower than $y = x^{2}$ .

Example 2: a horizontal stretch

Describe $h(x) = f\left(\dfrac{1}{2}x\right)$ compared to $f(x) = x^{2}$ .

The multiplier is on the input — horizontal dilation

b = \dfrac{1}{2}

Compare to

1

0 < \dfrac{1}{2} < 1

Between

0

and

1

means a horizontal stretch — the graph is wider

Answer: Horizontal stretch; the parabola is wider than $y = x^{2}$ .

Example 3: a reflection

Describe $r(x) = -f(x)$ compared to $f(x) = x^{2}$ .

The multiplier is negative

a = -1

A negative

a

reflects the graph across the

x

-axis

Since

|-1| = 1

, the width does not change

Answer: Reflection across the $x$ -axis; the parabola opens downward with the same width.

Try one yourself

Common questions

Why does a bigger

a

make the parabola look narrower?

Because the outputs climb faster. On $y = x^{2}$ , moving from $x = 1$ to $x = 2$ takes $y$ from $1$ to $4$ . On $y = 2x^{2}$ it goes from $2$ to $8$ . The graph shoots up sooner, which reads as a skinnier U.

How is a dilation different from a translation?

A translation slides the graph to a new spot without changing its shape. A dilation keeps the vertex of the parent function in place but stretches or compresses the shape itself.

What does a negative

a

do?

It reflects the parabola across the $x$ -axis, so it opens downward. The absolute value of $a$ still controls width: $|a| > 1$ is narrower and $|a| < 1$ is wider, reflected or not.

Why do horizontal dilations work backwards?

The multiplier changes the input before squaring. In $f(2x)$ , plugging in $x = 1$ computes $f(2)$ — the function reaches its values twice as fast, squeezing the graph toward the $y$ -axis even though $2 > 1$ .

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