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Year 9 Maths Introduction to Quadratic Equations and Graphs

Sketching y=ax^2 with Dilations and Reflections

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Theory

In \(y = ax^2\) the number \(a\) controls the shape of the parabola. This Year 9 maths page shows how a larger or smaller \(a\) makes the parabola narrower or wider (a dilation from the \(x\)-axis), how a negative \(a\) reflects it in the \(x\)-axis, how to sketch \(y = ax^2\) from \(y = x^2\) and how to find \(a\) from a point on the graph.

Every parabola \(y = ax^2\) (with \(a \ne 0\)) has its turning point at the origin \((0, 0)\) and its axis of symmetry on the \(y\)-axis, \(x = 0\). Only the shape changes as \(a\) changes.

A dilation from the \(x\)-axis by factor \(a\) multiplies every \(y\)-value by \(a\). Points move away from the \(x\)-axis if \(a > 1\), so \(y = ax^2\) is narrower than \(y = x^2\). Points move towards the \(x\)-axis if \(0 < a < 1\), so the parabola is wider. This is also called a vertical dilation or stretch.

A reflection in the \(x\)-axis changes the sign of every \(y\)-value. \(y = -x^2\) is \(y = x^2\) flipped upside down: it is concave down and its turning point is a maximum.

If \(a\) is negative, both happen: a dilation by factor \(|a|\), the size of \(a\) without its sign, and a reflection in the \(x\)-axis. The sign of \(a\) gives the concavity; the size of \(a\) gives the width.

A good way to see this is to type \(y = ax^2\) into a graphing app with a slider for \(a\), then drag \(a\) from \(5\) down to \(-5\) and watch the curve.

Narrower and wider parabolasCompared with y equals x squared (dashed), y equals 2.5 x squared is narrower and y equals 0.4 x squared is wider. All three have their turning point at the origin. xy−22246 y = 2.5x2 (narrower) y = x2 y = 0.4x2 (wider)
A larger \(a\) gives a narrower parabola, a smaller positive \(a\) gives a wider one. All three pass through \((0, 0)\).
Reflection in the x-axisThe parabola y equals 1.5 x squared is concave up. Its reflection in the x-axis, y equals negative 1.5 x squared, is concave down. Both have turning point at the origin. xy−22−4−224 (2, 6) (2, −6) y = 1.5x2 y = −1.5x2
Changing the sign of \(a\) reflects the parabola in the \(x\)-axis: \((2, 6)\) becomes \((2, -6)\).

The point with \(x = 1\) on \(y = ax^2\) is \((1, a)\). For any point \((x, y)\) on the graph:

\[y = ax^2 \quad\Rightarrow\quad a = \dfrac{y}{x^2}\]
a=yx2
EquationEffect on \(y = x^2\)
\(y = ax^2,\ a > 1\)dilation by factor \(a\) from the \(x\)-axis: narrower
\(y = ax^2,\ 0 < a < 1\)dilation by factor \(a\) from the \(x\)-axis: wider
\(y = -x^2\)reflection in the \(x\)-axis
\(y = ax^2,\ a < 0\)dilation by factor \(|a|\) and reflection in the \(x\)-axis
Finding \(a\) from a point. If \(y = ax^2\) passes through \((3, 6)\), then \(6 = a \times 9\), so \(a = \dfrac{2}{3}\).
Function notation. If \(f(x) = 2x^2\), then \(y = -f(x) = -2x^2\) is the reflection in the \(x\)-axis, \(y = 3f(x) = 6x^2\) is a dilation by factor \(3\) from the \(x\)-axis, and \(y = f(x) - 1 = 2x^2 - 1\) is a move \(1\) unit down.
Reflection in the \(y\)-axis. Replace \(x\) with \(-x\). \(y = ax^2\) does not change, because it is symmetric about the \(y\)-axis. A parabola that has been moved sideways does change: \(y = (x - 3)^2\) becomes \(y = (-x - 3)^2 = (x + 3)^2\), and its axis moves from \(x = 3\) to \(x = -3\).
Reading \(y = ax^2 + bx + c\). The sign of \(a\) still gives the concavity, and \(c\) is the \(y\)-intercept. For \(y = -x^2 + 4x + 1\): \(a = -1\), so concave down, crossing the \(y\)-axis at \(1\).

How to sketch \(y = ax^2\)

  1. Concavity: \(a > 0\) concave up, \(a < 0\) concave down.
  2. Turning point \((0, 0)\) and axis of symmetry \(x = 0\).
  3. Table of values: take the \(y\)-values of \(y = x^2\) (\(4, 1, 0, 1, 4\) for \(x = -2\) to \(2\)) and multiply each by \(a\).
  4. Plot the points and draw a smooth curve, symmetric about the \(y\)-axis. Label it with its equation.
  5. Describe it from \(y = x^2\): a dilation by factor \(|a|\) from the \(x\)-axis, and a reflection in the \(x\)-axis if \(a < 0\).

Comparing widths

  1. Ignore the sign and compare the sizes \(|a|\).
  2. The larger \(|a|\), the narrower the parabola. \(y = -4x^2\) is narrower than \(y = 2x^2\).

With a translation as well

  1. Order: for \(y = a(x - h)^2 + k\), dilate (and reflect), then move \(h\) right and \(k\) up. \(y = \dfrac{1}{2}(x - 4)^2 + 1\) is a dilation by factor \(\dfrac{1}{2}\), then \(4\) right and \(1\) up, with turning point \((4, 1)\).
  2. The order can matter. Moving \(y = x^2\) up \(1\) and then dilating by factor \(4\) gives \(y = 4(x^2 + 1) = 4x^2 + 4\), turning point \((0, 4)\). Dilating first, then moving up \(1\), gives \(y = 4x^2 + 1\), turning point \((0, 1)\). A dilation done after a vertical move stretches that move too.
Example 1 — Describing the transformation
Describe the transformation from \(y = x^2\) to each parabola, and say whether it is concave up or concave down.
(i) \(y = 6x^2\)
(ii) \(y = -\dfrac{2}{3}x^2\)
Solution

(i) \(a = 6\). Every \(y\)-value is multiplied by \(6\), so the point \((1, 1)\) on \(y = x^2\) moves to \((1, 6)\).

It is a dilation by factor \(6\) from the \(x\)-axis. The parabola is narrower and concave up.

(ii) \(a = -\dfrac{2}{3}\). The size \(\dfrac{2}{3}\) is less than \(1\), and the sign is negative.

It is a dilation by factor \(\dfrac{2}{3}\) from the \(x\)-axis and a reflection in the \(x\)-axis. The parabola is wider and concave down.

y=−23x2

Check (ii) at \(x = 3\): \(y = -\dfrac{2}{3} \times 9 = -6\). The turning point \((0, 0)\) is now a maximum.

Example 2 — Sketching from a table
Sketch \(y = -\dfrac{3}{4}x^2\) using a table of values for \(x = -4, -2, 0, 2, 4\).
Solution

Square each \(x\)-value first, then multiply by \(-\dfrac{3}{4}\). Opposite \(x\)-values give the same \(y\).

\(x = \pm 4:\ y\)\(=\)\(-\tfrac{3}{4}(16) = -12\)
\(x = \pm 2:\ y\)\(=\)\(-\tfrac{3}{4}(4) = -3\)
\(x = 0:\ y\)\(=\)\(0\)

Points: \((-4, -12)\), \((-2, -3)\), \((0, 0)\), \((2, -3)\), \((4, -12)\).

\(a < 0\), so concave down with a maximum at \((0, 0)\). \(|a| = \dfrac{3}{4} < 1\), so it is wider than \(y = x^2\).

Sketch of y = negative three quarters x squaredA concave down parabola with turning point at the origin passing through (2, negative 3), (negative 2, negative 3), (4, negative 12) and (negative 4, negative 12). The graph of y equals x squared is dashed for comparison. xy−4−224−12−8−44 (2, −3) (4, −12)

From \(y = x^2\) (dashed): a dilation by factor \(\dfrac{3}{4}\) from the \(x\)-axis and a reflection in the \(x\)-axis.

Example 3 — Finding \(a\) and ordering by width
Parabolas \(P\), \(Q\) and \(R\) all have equations of the form \(y = ax^2\). \(P\) passes through \((2, 6)\), \(Q\) through \((3, -3)\) and \(R\) through \((-2, 1)\).
(i) Find \(a\) for each parabola.
(ii) List them from narrowest to widest.
Solution

(i) Substitute each point into \(y = ax^2\).

\(P:\ 6\)\(=\)\(4a\)
\(a\)\(=\)\(\boldsymbol{\tfrac{3}{2}}\)
\(Q:\ -3\)\(=\)\(9a\)
\(a\)\(=\)\(\boldsymbol{-\tfrac{1}{3}}\)
\(R:\ 1\)\(=\)\(4a\)
\(a\)\(=\)\(\boldsymbol{\tfrac{1}{4}}\)
a=32,−13,14

(ii) Compare the sizes: \(\tfrac{3}{2} > \tfrac{1}{3} > \tfrac{1}{4}\). The sign of \(Q\) only makes it concave down.

From narrowest to widest: \(P\), \(Q\), \(R\).

Example 4 — Modelling an archway
A garden archway is modelled by \(y = ax^2\), with the top of the arch at the origin and \(x\) and \(y\) in metres. The arch is \(2.4\) m wide at its base, which is \(1.8\) m below the top.
(i) Find \(a\).
(ii) How far below the top is the arch at \(0.8\) m from the centre?
Solution

(i) The arch is concave down, so expect \(a < 0\). Half the width is \(1.2\) m, so the point \((1.2, -1.8)\) is on the curve.

\(-1.8\)\(=\)\(a \times 1.2^2\)
\(-1.8\)\(=\)\(1.44a\)
\(a\)\(=\)\(\boldsymbol{-1.25}\ \left(= -\tfrac{5}{4}\right)\)

(ii) Substitute \(x = 0.8\):

\(y\)\(=\)\(-1.25 \times 0.8^2\)
\(\)\(=\)\(-1.25 \times 0.64\)
\(\)\(=\)\(-0.8\)
y=−0.8

The arch is \(0.8\) m below the top there.

Common pitfalls

A bigger \(a\) is narrower, not wider. Multiplying the \(y\)-values by a larger number makes the curve climb faster, so it hugs the \(y\)-axis.
Width comes from the size of \(a\). \(y = -4x^2\) is narrower than \(y = 2x^2\), even though \(-4 < 2\). The minus sign only flips the graph.
Square before you multiply. For \(y = -\dfrac{1}{3}x^2\) at \(x = -3\): \((-3)^2 = 9\), then \(-\dfrac{1}{3} \times 9 = -3\). Squaring \(-\dfrac{1}{3}x\) instead gives a positive answer, which is wrong.
Know which axis. A reflection in the \(x\)-axis changes the sign of \(y\); a reflection in the \(y\)-axis changes the sign of \(x\). \(y = ax^2\) is unchanged by a reflection in the \(y\)-axis.

Frequently asked questions

What does \(a\) do in \(y = ax^2\)?

It sets the shape. A larger size of \(a\) makes the parabola narrower, a size between \(0\) and \(1\) makes it wider, and a negative \(a\) turns it upside down. The turning point stays at \((0, 0)\).

What is a dilation from the \(x\)-axis?

It multiplies every \(y\)-value by the same factor. Points move away from the \(x\)-axis if the factor is more than \(1\) and towards it if the factor is between \(0\) and \(1\). Points on the \(x\)-axis do not move.

How do you tell which parabola is narrower?

Compare the sizes of \(a\) and ignore the signs. The larger size gives the narrower parabola, so \(y = -5x^2\) is narrower than \(y = 2x^2\).

What happens when \(a\) is negative?

The parabola is reflected in the \(x\)-axis. It is concave down and its turning point is a maximum. It is also dilated by the size of \(a\).

How do you find \(a\) from a point on the graph?

Substitute the point into \(y = ax^2\) and solve for \(a\). For the point \((2, 10)\): \(10 = 4a\), so \(a = 2.5\).

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