Quadratic functions

Parabolas

1. What is a parabola?

A parabola is a curved graph with a very special shape: it goes down and then up, or up and then down.

You can find parabolas in real life:

the path of a ball when you throw it,
the shape of some bridges,
the shape of satellite dishes,
the water coming out of a fountain,
the path of a basketball shot.

In mathematics, parabolas usually appear when we study quadratic functions.

A quadratic function is a function whose formula includes an x2x^2x2 term.

For example:

y=x2y=x^2y=x2 y=2x2y=2x^2y=2x2 y=x2+3y=x^2+3y=x2+3 y=−x2+4y=-x^2+4y=−x2+4

2. The simplest parabola: y=x2y=x^2y=x2

Let’s make a table of values for:

y=x2y=x^2y=x2

xxx

y=x2y=x^2y=x2

-3

-2

-1

If we represent these points on a graph, we get a parabola.

The graph of y=x2y=x^2y=x2:

opens upwards,
has its lowest point at (0,0)(0,0)(0,0),
is symmetric,
has a U-shape.

The point (0,0)(0,0)(0,0) is called the vertex.

3. The vertex

The vertex is the turning point of the parabola.

It can be:

the lowest point, if the parabola opens upwards;
the highest point, if the parabola opens downwards.

Examples:

y=x2y=x^2y=x2

The vertex is:

(0,0)(0,0)(0,0) y=x2+2y=x^2+2y=x2+2

The vertex is:

(0,2)(0,2)(0,2) y=−x2+5y=-x^2+5y=−x2+5

The vertex is:

(0,5)(0,5)(0,5)

4. Parabolas opening upwards or downwards

Look at these two functions:

y=x2y=x^2y=x2 y=−x2y=-x^2y=−x2

The first one opens upwards.

The second one opens downwards.

The sign of the x2x^2x2 term is very important:

Function

Shape

y=x2y=x^2y=x2

Opens upwards

y=−x2y=-x^2y=−x2

Opens downwards

So:

if the coefficient of x2x^2x2 is positive, the parabola opens upwards;
if the coefficient of x2x^2x2 is negative, the parabola opens downwards.

Examples:

y=3x2y=3x^2y=3x2

opens upwards.

y=−2x2y=-2x^2y=−2x2

opens downwards.

5. Wider and narrower parabolas

Compare these functions:

y=x2y=x^2y=x2 y=2x2y=2x^2y=2x2 y=4x2y=4x^2y=4x2

The larger the number multiplying x2x^2x2, the narrower the parabola becomes.

Now compare:

y=x2y=x^2y=x2 y=0.5x2y=0.5x^2y=0.5x2 y=0.25x2y=0.25x^2y=0.25x2

The smaller the number multiplying x2x^2x2, the wider the parabola becomes.

Function

Shape

y=4x2y=4x^2y=4x2

Narrower

y=x2y=x^2y=x2

Normal

y=0.25x2y=0.25x^2y=0.25x2

Wider

6. Moving the parabola up and down

Look at these functions:

y=x2y=x^2y=x2 y=x2+3y=x^2+3y=x2+3 y=x2−2y=x^2-2y=x2−2

The number added at the end moves the parabola vertically.

Function

Movement

y=x2+3y=x^2+3y=x2+3

Moves 3 units up

y=x2−2y=x^2-2y=x2−2

Moves 2 units down

So:

y=x2+ky=x^2+ky=x2+k

moves the parabola:

up if k>0k>0k>0,
down if k<0k<0k<0.

Examples:

y=x2+5y=x^2+5y=x2+5

The vertex is (0,5)(0,5)(0,5).

y=x2−4y=x^2-4y=x2−4

The vertex is (0,−4)(0,-4)(0,−4).

7. Axis of symmetry

A parabola is symmetric. This means that one half is a mirror image of the other half.

For the function:

y=x2y=x^2y=x2

the axis of symmetry is the vertical line:

x=0x=0x=0

This line passes through the vertex.

For these functions:

y=x2+3y=x^2+3y=x2+3 y=x2−5y=x^2-5y=x2−5

the axis of symmetry is still:

x=0x=0x=0

because the parabola has only moved up or down.

8. Intercepts

Y-intercept

The y-intercept is the point where the graph crosses the y-axis.

To find it, we use:

x=0x=0x=0

Example:

y=x2+2y=x^2+2y=x2+2

If x=0x=0x=0:

y=02+2=2y=0^2+2=2y=02+2=2

So the y-intercept is:

(0,2)(0,2)(0,2)

X-intercepts

The x-intercepts are the points where the graph crosses the x-axis.

To find them, we use:

y=0y=0y=0

Example:

y=x2−4y=x^2-4y=x2−4

We solve:

0=x2−40=x^2-40=x2−4 x2=4x^2=4x2=4 x=2orx=−2x=2 \quad \text{or} \quad x=-2x=2orx=−2

So the x-intercepts are:

(−2,0)(-2,0)(−2,0)

and

(2,0)(2,0)(2,0)

Some parabolas have:

two x-intercepts,
one x-intercept,
no x-intercepts.

Examples:

y=x2−4y=x^2-4y=x2−4

has two x-intercepts.

y=x2y=x^2y=x2

has one x-intercept.

y=x2+4y=x^2+4y=x2+4

has no x-intercepts.

9. Increasing and decreasing

A parabola can be increasing or decreasing depending on the part of the graph.

For:

y=x2y=x^2y=x2

the parabola:

is decreasing when x<0x<0x<0,
reaches its minimum at x=0x=0x=0,
is increasing when x>0x>0x>0.

In simple words:

The graph goes down until it reaches the vertex, and then it goes up.

For:

y=−x2y=-x^2y=−x2

the opposite happens:

the graph goes up until it reaches the vertex,
then it goes down.

10. Maximum and minimum

A parabola can have a minimum or a maximum.

If it opens upwards, it has a minimum.

Example:

y=x2+1y=x^2+1y=x2+1

The vertex is (0,1)(0,1)(0,1), so the minimum value is:

y=1y=1y=1

If it opens downwards, it has a maximum.

Example:

y=−x2+4y=-x^2+4y=−x2+4

The vertex is (0,4)(0,4)(0,4), so the maximum value is:

y=4y=4y=4

11. Summary

A parabola is the graph of a quadratic function.

Important ideas:

Concept

Meaning

Vertex

The turning point of the parabola

Axis of symmetry

The vertical line that divides the parabola into two equal halves

Opens upwards

The parabola has a minimum

Opens downwards

The parabola has a maximum

Y-intercept

Point where the graph crosses the y-axis

X-intercepts

Points where the graph crosses the x-axis

12. Quick examples

Example 1

Study the function:

y=x2−3y=x^2-3y=x2−3

The parabola opens upwards because the coefficient of x2x^2x2 is positive.

The vertex is:

(0,−3)(0,-3)(0,−3)

The y-intercept is:

(0,−3)(0,-3)(0,−3)

To find the x-intercepts:

0=x2−30=x^2-30=x2−3 x2=3x^2=3x2=3

So the graph crosses the x-axis approximately at:

x=−1.7x=-1.7x=−1.7

and

x=1.7x=1.7x=1.7

Example 2

Study the function:

y=−x2+5y=-x^2+5y=−x2+5

The parabola opens downwards because the coefficient of x2x^2x2 is negative.

The vertex is:

(0,5)(0,5)(0,5)

The maximum value is:

y=5y=5y=5

The y-intercept is:

(0,5)(0,5)(0,5)

To find the x-intercepts:

0=−x2+50=-x^2+50=−x2+5 x2=5x^2=5x2=5

So the graph crosses the x-axis approximately at:

x=−2.2x=-2.2x=−2.2

and

x=2.2x=2.2x=2.2

13. Exercises

Exercise 1

Complete the table for:

y=x2−1y=x^2-1y=x2−1

xxx

y=x2−1y=x^2-1y=x2−1

-3

-2

-1

Then draw the graph.

Exercise 2

For each function, say whether the parabola opens upwards or downwards.

y=x2+2y=x^2+2y=x2+2

y=−x2+3y=-x^2+3y=−x2+3

y=4x2y=4x^2y=4x2

y=−0.5x2−1y=-0.5x^2-1y=−0.5x2−1

Exercise 3

Find the vertex of each parabola.

y=x2+6y=x^2+6y=x2+6

y=x2−4y=x^2-4y=x2−4

y=−x2+2y=-x^2+2y=−x2+2

y=−x2−3y=-x^2-3y=−x2−3

Exercise 4

Find the y-intercept of each function.

y=x2+7y=x^2+7y=x2+7

y=x2−5y=x^2-5y=x2−5

y=−x2+1y=-x^2+1y=−x2+1

y=−x2−4y=-x^2-4y=−x2−4

Exercise 5

Find the x-intercepts.

y=x2−9y=x^2-9y=x2−9

y=x2−1y=x^2-1y=x2−1

y=−x2+4y=-x^2+4y=−x2+4

y=x2+3y=x^2+3y=x2+3

14. Final challenge

A basketball player throws a ball. The height of the ball can be modelled by this function:

h=−x2+6xh=-x^2+6xh=−x2+6x

where:

xxx is the horizontal distance in metres,
hhh is the height of the ball in metres.

Answer the questions:

Does the parabola open upwards or downwards?
What does this tell us about the movement of the ball?
Complete the table:

xxx

h=−x2+6xh=-x^2+6xh=−x2+6x

What is the maximum height of the ball?
At what distance does the ball touch the ground again?

Key ideas to remember

A parabola is not just a curve. It tells a story.

It can show:

something going up and down,
a maximum height,
a minimum value,
symmetry,
movement,
growth and decrease.

The most important point of a parabola is the vertex, because it tells us where the graph changes direction.

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