Drawing Quadratic Graphs

What you'll learn

How to complete a table of values for a quadratic formula.
How to plot points and draw a smooth quadratic curve.
How to read roots and turning points from a graph.
How to use a graph to estimate solutions to equations.

1. Starting point: coordinates and substitution

A coordinate tells you a point’s position on a grid. It is written as $(x, y)$ , where the first number is across and the second number is up or down.

To draw a graph, you choose values of $x$ , substitute them into the formula, and calculate the matching values of $y$ .

Example

Substituting into a quadratic formula

For $y=x^2-2x-1$ , find $y$ when $x = - 1$ , $x = 0$ , and $x = 3$ .

Substitute $x = - 1$ :

$y=(-1)^2-2(-1)-1=1+2-1=2$
Substitute $x = 0$ :

$y=0^2-2(0)-1=-1$
Substitute $x = 3$ :

$y=3^2-2(3)-1=9-6-1=2$
So the points you would plot are $(- 1, 2)$ , $(0, - 1)$ , and $(3, 2)$ .

Common Mistake

Negative x-values

When you square a negative number, the answer is positive: $3)^2=9$ , not -9. Use brackets when substituting negative values.

2. What is a quadratic graph?

Definition

Quadratic graph

A quadratic graph is the graph of a formula with an $x^2$ term, such as $y=x^2+x-6$ or $y=7x-x^2$ . Its shape is a smooth curve called a parabola.

A parabola can open upwards or downwards.

Quadratic graphs are parabolas, opening upwards when the x² coefficient is positive and downwards when it is negative.

If the $x^2$ term is positive, the graph opens upwards like a smile.
If the $x^2$ term is negative, the graph opens downwards like a frown.

Key Idea

The overall shape

Quadratic graphs are not straight lines. They should be drawn as one smooth curve through the plotted points.

Example

Deciding the shape

Look at $y=5x-x^2$ . Decide whether the graph opens upwards or downwards.

The negative x² term makes this parabola open downwards, giving it a highest point.

Rewrite the formula mentally as $y=-x^2+5x$ .
The coefficient of $x^2$ is negative.
So the graph opens downwards.
This means it will have a highest point rather than a lowest point.

3. Completing a table of values

A table of values lists chosen $x$ -values and their matching $y$ -values. You use it to create points for your graph.

Example

Completing a table of values

Complete the table for $y=x^2-x-4$ , for $x = - 2$ to $x = 4$ .

The completed table gives the coordinates that will be plotted to draw the quadratic curve.

Substitute $x = - 2$ :

$y=(-2)^2-(-2)-4=4+2-4=2$
Substitute $x = - 1$ :

$y=(-1)^2-(-1)-4=1+1-4=-2$
Continue in the same way for each $x$ -value.
The completed table is:

x	-2	-1	0	1	2	3	4
y	2	-2	-4	-4	-2	2	8

Tip

Quick check

Quadratic tables often have a pattern: the $y$ -values decrease, reach a lowest or highest area, then increase again. If one value looks wildly out of place, recalculate it.

4. Plotting and drawing the graph

Once the table is complete, turn each pair into a coordinate. For example, if $x = - 2$ and $y = 2$ , plot $(- 2, 2)$ .

Use small crosses or dots, then draw a smooth curve through them.

Example

Drawing the graph from a table

Use these values for $y=x^2-x-4$ :

Plot the table values and draw one smooth U-shaped curve through the points.

x	-2	-1	0	1	2	3	4
y	2	-2	-4	-4	-2	2	8

Turn the table into points: $(- 2, 2)$ , $(- 1, - 2)$ , $(0, - 4)$ , $(1, - 4)$ , $(2, - 2)$ , $(3, 2)$ , and $(4, 8)$ .
Plot each point carefully on the grid.
Check that the points make a curved pattern, not a straight-line pattern.
Draw one smooth U-shaped curve through the points.
The lowest part of the graph is between $x = 0$ and $x = 1$ , so the turning point is about $(0.5, - 4.3)$ .

Common Mistake

Joining with straight lines

Do not join quadratic points using a ruler. A quadratic graph is a smooth curve, so the line should bend gradually through the points.

5. Roots and turning points

Definition

Roots and turning point

The roots of a quadratic are the $x$ -values where the graph crosses the $x$ -axis. The turning point is the lowest point or highest point of the parabola.

Roots are read at the x-axis crossings, while the turning point is the lowest or highest point of the curve.

Roots happen where $y = 0$ , because every point on the $x$ -axis has a $y$ -coordinate of 0.

Example

Reading from a quadratic graph

A graph of $y=x^2-4x+3$ has its lowest point at $(2, - 1)$ . It crosses the $x$ -axis at $x = 1$ and $x = 3$ . Find the turning point and the roots of $x^2-4x+3=0$ .

For y = x² − 4x + 3, the roots are x = 1 and x = 3 and the turning point is (2, −1).

The turning point is the lowest point of the graph.
So the turning point is $(2, - 1)$ .
The roots are where the graph crosses the $x$ -axis.
The graph crosses at $x = 1$ and $x = 3$ .
So the roots are $x = 1$ and $x = 3$ .

6. Solving equations using a quadratic graph

Sometimes the equation is not equal to 0. For example, to solve $x^2-x-4=-2$ , you look for where the graph has $y = - 2$ .

That means you use a horizontal line, which goes straight across the grid.

Example

Solving an equation from a graph

Use the graph of $y=x^2-x-4$ to solve $x^2-x-4=-2$ .

The solutions are the x-values where the curve meets the horizontal line y = −2.

Notice that $x^2-x-4$ is the expression used for $y$ .
The equation asks where $y = - 2$ .
On the graph, draw or imagine the horizontal line $y = - 2$ .
Read the $x$ -values where this horizontal line meets the curve.
From the table and graph, this happens at $x = - 1$ and $x = 2$ .

Sometimes you need to rearrange the equation first.

Example

Rearranging before using the graph

Use the graph of $y=x^2-3x-4$ to solve $x^2=3x+4$ .

After rearranging to x² − 3x − 4 = 0, the solutions are read from the roots of the graph.

Rearrange the equation so one side is 0:

$x^2-3x-4=0$
This matches the graph $y=x^2-3x-4$ .
So you need the roots of the graph.
Read the $x$ -axis crossings.
If the graph crosses at $x = - 1$ and $x = 4$ , the solutions are $x = - 1$ and $x = 4$ .

Common Mistake

Forgetting to rearrange

If the equation is $x^2=3x+4$ , do not look for where $y = 3 x + 4$ on a graph of $y=x^2-3x-4$ . Rearrange first, then use the roots.

Exam technique

In the exam

Complete the table carefully, especially negative $x$ -values and any minus sign in front of $x^2$ .
Plot points accurately, then draw a smooth curve rather than straight line segments.
Roots are $x$ -axis crossings; equations like $x^2-x-4=-2$ use the horizontal line $y = - 2$ .

Self review

Check yourself

If $x = - 3$ , what is $x^2$ ?
Where on a quadratic graph do you read the roots?
How would you use a graph of $y=x^2-2x-1$ to solve $x^2-2x-1=3$ ?

Drawing Quadratic Graphs

What you'll learn

How to complete a table of values for a quadratic formula.
How to plot points and draw a smooth quadratic curve.
How to read roots and turning points from a graph.
How to use a graph to estimate solutions to equations.

1. Starting point: coordinates and substitution

A coordinate tells you a point’s position on a grid. It is written as $(x, y)$ , where the first number is across and the second number is up or down.

To draw a graph, you choose values of $x$ , substitute them into the formula, and calculate the matching values of $y$ .

Example

Substituting into a quadratic formula

For $y=x^2-2x-1$ , find $y$ when $x = - 1$ , $x = 0$ , and $x = 3$ .

Substitute $x = - 1$ :

$y=(-1)^2-2(-1)-1=1+2-1=2$
Substitute $x = 0$ :

$y=0^2-2(0)-1=-1$
Substitute $x = 3$ :

$y=3^2-2(3)-1=9-6-1=2$
So the points you would plot are $(- 1, 2)$ , $(0, - 1)$ , and $(3, 2)$ .

Common Mistake

Negative x-values

When you square a negative number, the answer is positive: $3)^2=9$ , not -9. Use brackets when substituting negative values.

2. What is a quadratic graph?

Definition

Quadratic graph

A quadratic graph is the graph of a formula with an $x^2$ term, such as $y=x^2+x-6$ or $y=7x-x^2$ . Its shape is a smooth curve called a parabola.

A parabola can open upwards or downwards.

Quadratic graphs are parabolas, opening upwards when the x² coefficient is positive and downwards when it is negative.

If the $x^2$ term is positive, the graph opens upwards like a smile.
If the $x^2$ term is negative, the graph opens downwards like a frown.

Key Idea

The overall shape

Quadratic graphs are not straight lines. They should be drawn as one smooth curve through the plotted points.

Example

Deciding the shape

Look at $y=5x-x^2$ . Decide whether the graph opens upwards or downwards.

The negative x² term makes this parabola open downwards, giving it a highest point.

Rewrite the formula mentally as $y=-x^2+5x$ .
The coefficient of $x^2$ is negative.
So the graph opens downwards.
This means it will have a highest point rather than a lowest point.

3. Completing a table of values

A table of values lists chosen $x$ -values and their matching $y$ -values. You use it to create points for your graph.

Example

Completing a table of values

Complete the table for $y=x^2-x-4$ , for $x = - 2$ to $x = 4$ .

The completed table gives the coordinates that will be plotted to draw the quadratic curve.

Substitute $x = - 2$ :

$y=(-2)^2-(-2)-4=4+2-4=2$
Substitute $x = - 1$ :

$y=(-1)^2-(-1)-4=1+1-4=-2$
Continue in the same way for each $x$ -value.
The completed table is:

x	-2	-1	0	1	2	3	4
y	2	-2	-4	-4	-2	2	8

Tip

Quick check

Quadratic tables often have a pattern: the $y$ -values decrease, reach a lowest or highest area, then increase again. If one value looks wildly out of place, recalculate it.

4. Plotting and drawing the graph

Once the table is complete, turn each pair into a coordinate. For example, if $x = - 2$ and $y = 2$ , plot $(- 2, 2)$ .

Use small crosses or dots, then draw a smooth curve through them.

Example

Drawing the graph from a table

Use these values for $y=x^2-x-4$ :

Plot the table values and draw one smooth U-shaped curve through the points.

x	-2	-1	0	1	2	3	4
y	2	-2	-4	-4	-2	2	8

Turn the table into points: $(- 2, 2)$ , $(- 1, - 2)$ , $(0, - 4)$ , $(1, - 4)$ , $(2, - 2)$ , $(3, 2)$ , and $(4, 8)$ .
Plot each point carefully on the grid.
Check that the points make a curved pattern, not a straight-line pattern.
Draw one smooth U-shaped curve through the points.
The lowest part of the graph is between $x = 0$ and $x = 1$ , so the turning point is about $(0.5, - 4.3)$ .

Common Mistake

Joining with straight lines

Do not join quadratic points using a ruler. A quadratic graph is a smooth curve, so the line should bend gradually through the points.

5. Roots and turning points

Definition

Roots and turning point

The roots of a quadratic are the $x$ -values where the graph crosses the $x$ -axis. The turning point is the lowest point or highest point of the parabola.

Roots are read at the x-axis crossings, while the turning point is the lowest or highest point of the curve.

Roots happen where $y = 0$ , because every point on the $x$ -axis has a $y$ -coordinate of 0.

Example

Reading from a quadratic graph

A graph of $y=x^2-4x+3$ has its lowest point at $(2, - 1)$ . It crosses the $x$ -axis at $x = 1$ and $x = 3$ . Find the turning point and the roots of $x^2-4x+3=0$ .

For y = x² − 4x + 3, the roots are x = 1 and x = 3 and the turning point is (2, −1).

The turning point is the lowest point of the graph.
So the turning point is $(2, - 1)$ .
The roots are where the graph crosses the $x$ -axis.
The graph crosses at $x = 1$ and $x = 3$ .
So the roots are $x = 1$ and $x = 3$ .

6. Solving equations using a quadratic graph

Sometimes the equation is not equal to 0. For example, to solve $x^2-x-4=-2$ , you look for where the graph has $y = - 2$ .

That means you use a horizontal line, which goes straight across the grid.

Example

Solving an equation from a graph

Use the graph of $y=x^2-x-4$ to solve $x^2-x-4=-2$ .

The solutions are the x-values where the curve meets the horizontal line y = −2.

Notice that $x^2-x-4$ is the expression used for $y$ .
The equation asks where $y = - 2$ .
On the graph, draw or imagine the horizontal line $y = - 2$ .
Read the $x$ -values where this horizontal line meets the curve.
From the table and graph, this happens at $x = - 1$ and $x = 2$ .

Sometimes you need to rearrange the equation first.

Example

Rearranging before using the graph

Use the graph of $y=x^2-3x-4$ to solve $x^2=3x+4$ .

After rearranging to x² − 3x − 4 = 0, the solutions are read from the roots of the graph.

Rearrange the equation so one side is 0:

$x^2-3x-4=0$
This matches the graph $y=x^2-3x-4$ .
So you need the roots of the graph.
Read the $x$ -axis crossings.
If the graph crosses at $x = - 1$ and $x = 4$ , the solutions are $x = - 1$ and $x = 4$ .

Common Mistake

Forgetting to rearrange

If the equation is $x^2=3x+4$ , do not look for where $y = 3 x + 4$ on a graph of $y=x^2-3x-4$ . Rearrange first, then use the roots.

Exam technique

In the exam

Complete the table carefully, especially negative $x$ -values and any minus sign in front of $x^2$ .
Plot points accurately, then draw a smooth curve rather than straight line segments.
Roots are $x$ -axis crossings; equations like $x^2-x-4=-2$ use the horizontal line $y = - 2$ .

Self review

Check yourself

If $x = - 3$ , what is $x^2$ ?
Where on a quadratic graph do you read the roots?
How would you use a graph of $y=x^2-2x-1$ to solve $x^2-2x-1=3$ ?

Drawing Quadratic Graphs

What you'll learn

1. Starting point: coordinates and substitution

2. What is a quadratic graph?

3. Completing a table of values

4. Plotting and drawing the graph

5. Roots and turning points

6. Solving equations using a quadratic graph

Recap questions

You've reached the end

Drawing Quadratic Graphs

What you'll learn

1. Starting point: coordinates and substitution

2. What is a quadratic graph?

3. Completing a table of values

4. Plotting and drawing the graph

5. Roots and turning points

6. Solving equations using a quadratic graph

Recap questions

You've reached the end

Drawing Quadratic Graphs

What you'll learn

1. Starting point: coordinates and substitution

2. What is a quadratic graph?

3. Completing a table of values

4. Plotting and drawing the graph

5. Roots and turning points

6. Solving equations using a quadratic graph

Recap questions

You've reached the end

Drawing Quadratic Graphs Revision Guide

Drawing Quadratic Graphs

What you'll learn

1. Starting point: coordinates and substitution

2. What is a quadratic graph?

3. Completing a table of values

4. Plotting and drawing the graph

5. Roots and turning points

6. Solving equations using a quadratic graph

Recap questions

You've reached the end

Drawing Quadratic Graphs Revision Guide