Year 9: Parabolas

Back to main page

What is a parabola

In year 9, you begin working with curved graphs as well as straight lines. One of the most important curved graphs in mathematics is the parabola.

A parabola is the graph of any equation that includes $x ²$ , such as:

$y = x ²$
$y = x ² + 3$
$y = - 2 x ² + 4 x - 1$

In general, a quadratic equation is written as:

$y = a x ² + b x + c$

Here, $a$ , $b$ and $c$ are numbers, and $a$ cannot be zero.

Basic shape of parabolas

Every parabola has a U shape. Depending on the value of $a$ , the parabola can:

Open upwards if $a > 0$ .
Open downwards if $a < 0$ .

Example 1: Y = x²

Consider the equation $y = x ²$ .

Table of values:

If $x = - 2$ , then $y = 4$ .
If $x = - 1$ , then $y = 1$ .
If $x = 0$ , then $y = 0$ .
If $x = 1$ , then $y = 1$ .
If $x = 2$ , then $y = 4$ .

When plotted and connected smoothly, these points form a U-shaped curve opening upwards.

Example 2: Y = −x²

Now consider $y = - x ²$ .

Table of values:

If $x = - 2$ , then $y = - 4$ .
If $x = - 1$ , then $y = - 1$ .
If $x = 0$ , then $y = 0$ .
If $x = 1$ , then $y = - 1$ .
If $x = 2$ , then $y = - 4$ .

This forms a U-shaped curve opening downwards.

Key idea: The sign of

a

tells you whether the parabola opens up or down.

Vertex and axis of symmetry

Every parabola has a special point called the vertex, which is its minimum or maximum point.

If the parabola opens upwards, the vertex is the lowest point.
If it opens downwards, the vertex is the highest point.

Every parabola also has an axis of symmetry—a vertical line that divides it into two mirrored halves.

Example 3: Vertex of y = x²

For $y = x ²$ , the smallest value of $y$ is 0 at $x = 0$ .

The vertex is at (0, 0).
The axis of symmetry is x = 0.

Example 4: Vertex of y = x² + 4

Adding 4 to $x ²$ shifts the graph up by 4 units.

The vertex of $y = x ²$ is (0, 0).
The vertex of $y = x ² + 4$ is (0, 4).

The axis of symmetry remains x = 0.

Standard form of a parabola

A quadratic equation in standard form looks like:

$y = a x ² + b x + c$

From this form, you can:

Determine the direction of opening from $a$ .
Estimate the location of the vertex.
Find intercepts.

Axis of symmetry formula (extension): For

y = a x ² + b x + c

, the axis of symmetry is

x = - b \div (2 a)

Finding y intercepts

The y intercept is the point where the parabola crosses the y axis. On the y axis, $x = 0$ .

To find the y intercept, substitute $x = 0$ into the equation.

Example 5: Y intercept of y = x² − 3x + 2

Set $x = 0$ :

$y = 0 - 0 + 2 = 2$

The y intercept is (0, 2).

Example 6: Y intercept of y = −2x² + 4x − 1

Set $x = 0$ :

$y = - 1$

The y intercept is (0, −1).

Finding x intercepts

The x intercepts are where the parabola crosses the x axis. On the x axis, $y = 0$ .

To find x intercepts, solve $0 = a x ² + b x + c$ .

A parabola can have:

Two x intercepts,
One x intercept (touching the axis), or
No x intercepts.

Example 7: X intercepts of y = x² − 3x + 2

Solve $0 = x ² - 3 x + 2$ :

$0 = (x - 1) (x - 2)$

Thus, $x = 1$ or $x = 2$ .

The x intercepts are (1, 0) and (2, 0).

Example 8: X intercept of y = (x − 3)²

Solve $0 = (x - 3) ²$ :

The only solution is $x = 3$ .

The parabola touches the x axis at (3, 0).

Example 9: No real x intercepts

Consider $y = x ² + 4$ .

Setting $y = 0$ gives $x ² = - 4$ , which has no real solutions.

This parabola does not cross the x axis.

Sketching parabolas using key points

To sketch a parabola quickly, you need:

The vertex,
The y intercept,
The x intercepts (if they exist).

Example 10: Sketch y = x² − 4

Step 1: Find the vertex. This is $y = x ²$ shifted down 4 units, so the vertex is (0, −4).

Step 2: Find the y intercept. Substitute $x = 0$ , giving (0, −4).

Step 3: Find the x intercepts.

Solve $0 = x ² - 4$ , giving $x = \pm 2$ .

The x intercepts are (−2, 0) and (2, 0).

Step 4: Sketch. Plot the vertex and intercepts, then draw a smooth U shape.

Effect of the number a

The value of $a$ affects the steepness or width of the parabola.

If |a| is small (e.g., 0.5), the parabola is wide.
If |a| is large (e.g., 3), the parabola is narrow and steep.

Example 11: Comparing y = x² and y = 3x²

Both open upwards, but $y = 3 x ²$ is narrower.

Example 12: Comparing y = −x² and y = −0.5x²

Both open downwards, but $y = - 0.5 x ²$ is wider.

Parabolas in worded problems

Parabolas often appear in real life, especially when something is thrown or falling under gravity.

Example 13: Path of a thrown ball

A ball is thrown upwards and its height after t seconds is:

$h = - 5 t ² + 20 t$

Starting height (y intercept) is (0, 0).

When the ball hits the ground Solve $0 = - 5 t ² + 20 t$ to get t = 0 or t = 4.

The ball returns to the ground after 4 seconds.

Maximum height (vertex) Halfway between 0 and 4 is t = 2.

Substituting t = 2 gives 20 metres.

Example 14: Area of a rectangle

A rectangle has a fixed perimeter of 20 cm. If the length is x, the width is $10 - x$ .

The area is $A = - x ² + 10 x$ , a downward opening parabola.

X intercepts are x = 0 and x = 10.

Maximum area occurs when x = 5, making the rectangle a square.

Summary

A parabola is the graph of a quadratic equation.
The sign of $a$ shows whether it opens up or down.
Every parabola has a vertex and an axis of symmetry.
Y intercepts occur where x = 0.
X intercepts occur where y = 0.
Parabolas appear in many real situations involving objects in motion or maximum/minimum values.