Drawing Quadratic Graphs

What you'll learn

How to recognise a quadratic graph and its curved shape.
How to complete a table of values by substituting different x-values.
How to plot the coordinates accurately and join them with a smooth curve.
How to check your graph using intercepts, symmetry and the turning point.

The basic idea

A straight-line graph comes from equations like $y = 2 x + 1$ . A quadratic graph comes from an equation containing an $x^2$ term, such as $y=x^2-2x-3$ .

Definition

Quadratic graph

A quadratic graph is the graph of an equation where the highest power of $x$ is $x^2$ . Its shape is called a parabola, which is a smooth U-shaped curve or upside-down U-shaped curve.

Quadratic graphs are not drawn with a ruler through the points. They are curved, so you draw a smooth curve passing through the plotted points.

Key Idea

Main takeaway

To draw a quadratic graph, you usually make a table of values, plot the coordinates, then draw a smooth parabola through the points.

Prerequisite: coordinates

A coordinate tells you the position of a point on a graph. It is written as an ordered pair, like $(2, 5)$ .

The first number is the x-coordinate, which tells you how far left or right to move. The second number is the y-coordinate, which tells you how far up or down to move.

Example

Plotting a coordinate

Plot the point $(3, - 2)$ .

Start at the origin, which is the point $(0, 0)$ .
Move 3 squares to the right because the x-coordinate is positive 3.
Move 2 squares down because the y-coordinate is negative 2.
Mark the point with a small cross and label it if needed.

Common Mistake

Swapping the coordinates

The point $(3, - 2)$ is different from $(- 2, 3)$ . Always move along the x-axis first, then up or down for the y-coordinate.

Substituting into a quadratic

To draw the graph, you need several coordinates. You get these by choosing x-values and substituting them into the equation to find the matching y-values.

Definition

Substitution

Substitution means replacing a variable with a number. For example, if $y=x^2+1$ and $x = 3$ , then you replace $x$ with 3.

Be especially careful when substituting negative numbers. A negative number squared becomes positive, so $2)^2=4$ .

Example

Finding one y-value

For the equation $y=x^2-2x-3$ , find $y$ when $x = - 2$ .

Write the equation and replace $x$ with -2:
$y=(-2)^2-2(-2)-3$
Work out each part. Remember that $2)^2=4$ and $- 2 (- 2) = 4$ .
Add the values together:
$y = 4 + 4 - 3 = 5$
So when $x = - 2$ , the coordinate is $(- 2, 5)$ .

Common Mistake

Forgetting brackets with negatives

If $x = - 2$ , then $x^2$ means $2)^2$ , not $2^2$ . Use brackets when substituting negative values into squared terms.

Completing a table of values

A table of values is a table showing several x-values and their matching y-values. It gives you the coordinates you need to plot.

You are often given the x-values. Your job is to fill in the missing y-values by substituting each x-value into the equation.

Example

Completing a table for a quadratic

Complete the table for $y=x^2-2x-3$ .

x	-2	-1	0	1	2	3	4
y

Substitute $x = - 2$ :
$y=(-2)^2-2(-2)-3=5$
Substitute $x = - 1$ :
$y=(-1)^2-2(-1)-3=0$
Substitute $x = 0$ :
$y=0^2-2(0)-3=-3$
Substitute $x = 1$ :
$y=1^2-2(1)-3=-4$
Continue in the same way for $x = 2$ , $x = 3$ and $x = 4$ .
Fill in the completed table:

x	-2	-1	0	1	2	3	4
y	5	0	-3	-4	-3	0	5

Tip

Use the shape to check the table

Quadratic tables often have a pattern. In this example the y-values go 5, 0, -3, -4, -3, 0, 5, which is symmetric around the lowest value.

Plotting the points

Once your table is complete, turn each column into a coordinate:

x = -2, y = 5 gives $(- 2, 5)$
x = -1, y = 0 gives $(- 1, 0)$
x = 0, y = -3 gives $(0, - 3)$
x = 1, y = -4 gives $(1, - 4)$
x = 2, y = -3 gives $(2, - 3)$
x = 3, y = 0 gives $(3, 0)$
x = 4, y = 5 gives $(4, 5)$

Example

Using a table to list coordinates

Use the completed table to write down the coordinates for the graph of $y=x^2-2x-3$ .

Match each x-value with the y-value directly underneath it.
Write each pair in the form $(x, y)$ .
The coordinates are:
$(4,5)(-2,5),\ (-1,0),\ (0,-3),\ (1,-4),\ (2,-3),\ (3,0),\ (4,5)$
Plot each point carefully on the coordinate grid.

Common Mistake

Joining points too early

Do not draw the curve until all the points have been plotted. One wrong point can make the whole curve look distorted.

Drawing the smooth curve

After plotting the points, draw a smooth curve through them. For $y=x^2-2x-3$ , the coefficient of $x^2$ is positive, so the parabola opens upwards.

The graph below shows the completed curve for this example, including the intercepts and turning point.

A labelled parabola for y equals x squared minus 2x minus 3, showing plotted points, x-intercepts, y-intercept and turning point.

Definition

Turning point and intercepts

The turning point is the point where the parabola changes direction. An x-intercept is where the graph crosses the x-axis. A y-intercept is where the graph crosses the y-axis.

Example

Drawing the graph from the points

Draw the graph of $y=x^2-2x-3$ using the completed table.

Draw a pair of axes with a suitable scale. The x-values go from -2 to 4, and the y-values go from -4 to 5, so make sure your grid includes these values.
Plot the coordinates:
$(4,5)(-2,5),\ (-1,0),\ (0,-3),\ (1,-4),\ (2,-3),\ (3,0),\ (4,5)$
Check that the points form a smooth U-shape rather than a straight line.
Draw a smooth curve through the points, making the bottom of the curve near $(1, - 4)$ .
Extend the curve slightly beyond the end points if the grid allows, but do not invent a completely different shape.

Tip

Pencil first

In an exam, draw the curve lightly in pencil first. If one point looks out of place, recheck its substitution before committing to the curve.

How to check your quadratic graph

There are three useful checks.

1. Check the y-intercept

The y-intercept happens when $x = 0$ . For $y=x^2-2x-3$ , substituting $x = 0$ gives $y = - 3$ , so the graph should cross the y-axis at $(0, - 3)$ .

2. Check the x-intercepts

The x-intercepts happen when $y = 0$ . From the table, $y = 0$ when x is -1 and 3, so the graph should cross the x-axis at $(- 1, 0)$ and $(3, 0)$ .

3. Check the turning point

The turning point should sit in the middle of the symmetry. Here the matching y-values are:

5 and 5
0 and 0
-3 and -3

So the lowest point is halfway between the two sides, at about $(1, - 4)$ .

Example

Spotting an error in a plotted point

A student draws $y=x^2-2x-3$ but plots the point for $x = 2$ at $(2, 3)$ . Explain why this is likely to be wrong.

Substitute $x = 2$ into the equation:
$y=2^2-2(2)-3$
Simplify the value:
$y = 4 - 4 - 3 = - 3$
The correct point is $(2, - 3)$ , not $(2, 3)$ .
Also, the table has symmetry: the points $(0, - 3)$ and $(2, - 3)$ should have the same y-value.

Common Mistake

Do not force a straight line

Even if the plotted points look nearly straight over a small section, a quadratic graph must be drawn as a smooth curve, not as joined straight-line segments.

What if the parabola opens downwards?

If the coefficient of $x^2$ is negative, the parabola opens downwards instead of upwards. For example, $y=-x^2+4$ has an upside-down U-shape.

Example

A downward quadratic

Complete a few values for $y=-x^2+4$ .

Substitute $x = - 2$ :
$y=-(-2)^2+4=-4+4=0$
Substitute $x = 0$ :
$y=-(0)^2+4=4$
Substitute $x = 2$ :
$y=-(2)^2+4=0$
The points $(- 2, 0)$ , $(0, 4)$ and $(2, 0)$ show that the graph has a highest point instead of a lowest point.

Exam technique

In the exam

Complete the table carefully, showing at least one substitution if working marks are available.
Plot every coordinate with a small, neat cross, then check for a smooth quadratic pattern.
Draw a single smooth curve through the points; do not use a ruler to join them with straight lines.

Self review

Check yourself

If $y=x^2+3x-4$ , what value of $y$ do you get when $x = - 1$ ?
Why should a quadratic graph be drawn as a curve rather than straight line segments?
How can the y-intercept help you check whether your graph is sensible?

Drawing Quadratic Graphs

What you'll learn

How to recognise a quadratic graph and its curved shape.
How to complete a table of values by substituting different x-values.
How to plot the coordinates accurately and join them with a smooth curve.
How to check your graph using intercepts, symmetry and the turning point.

The basic idea

A straight-line graph comes from equations like $y = 2 x + 1$ . A quadratic graph comes from an equation containing an $x^2$ term, such as $y=x^2-2x-3$ .

Definition

Quadratic graph

A quadratic graph is the graph of an equation where the highest power of $x$ is $x^2$ . Its shape is called a parabola, which is a smooth U-shaped curve or upside-down U-shaped curve.

Quadratic graphs are not drawn with a ruler through the points. They are curved, so you draw a smooth curve passing through the plotted points.

Key Idea

Main takeaway

To draw a quadratic graph, you usually make a table of values, plot the coordinates, then draw a smooth parabola through the points.

Prerequisite: coordinates

A coordinate tells you the position of a point on a graph. It is written as an ordered pair, like $(2, 5)$ .

The first number is the x-coordinate, which tells you how far left or right to move. The second number is the y-coordinate, which tells you how far up or down to move.

Example

Plotting a coordinate

Plot the point $(3, - 2)$ .

Start at the origin, which is the point $(0, 0)$ .
Move 3 squares to the right because the x-coordinate is positive 3.
Move 2 squares down because the y-coordinate is negative 2.
Mark the point with a small cross and label it if needed.

Common Mistake

Swapping the coordinates

The point $(3, - 2)$ is different from $(- 2, 3)$ . Always move along the x-axis first, then up or down for the y-coordinate.

Substituting into a quadratic

To draw the graph, you need several coordinates. You get these by choosing x-values and substituting them into the equation to find the matching y-values.

Definition

Substitution

Substitution means replacing a variable with a number. For example, if $y=x^2+1$ and $x = 3$ , then you replace $x$ with 3.

Be especially careful when substituting negative numbers. A negative number squared becomes positive, so $2)^2=4$ .

Example

Finding one y-value

For the equation $y=x^2-2x-3$ , find $y$ when $x = - 2$ .

Write the equation and replace $x$ with -2:
$y=(-2)^2-2(-2)-3$
Work out each part. Remember that $2)^2=4$ and $- 2 (- 2) = 4$ .
Add the values together:
$y = 4 + 4 - 3 = 5$
So when $x = - 2$ , the coordinate is $(- 2, 5)$ .

Common Mistake

Forgetting brackets with negatives

If $x = - 2$ , then $x^2$ means $2)^2$ , not $2^2$ . Use brackets when substituting negative values into squared terms.

Completing a table of values

A table of values is a table showing several x-values and their matching y-values. It gives you the coordinates you need to plot.

You are often given the x-values. Your job is to fill in the missing y-values by substituting each x-value into the equation.

Example

Completing a table for a quadratic

Complete the table for $y=x^2-2x-3$ .

x	-2	-1	0	1	2	3	4
y

Substitute $x = - 2$ :
$y=(-2)^2-2(-2)-3=5$
Substitute $x = - 1$ :
$y=(-1)^2-2(-1)-3=0$
Substitute $x = 0$ :
$y=0^2-2(0)-3=-3$
Substitute $x = 1$ :
$y=1^2-2(1)-3=-4$
Continue in the same way for $x = 2$ , $x = 3$ and $x = 4$ .
Fill in the completed table:

x	-2	-1	0	1	2	3	4
y	5	0	-3	-4	-3	0	5

Tip

Use the shape to check the table

Quadratic tables often have a pattern. In this example the y-values go 5, 0, -3, -4, -3, 0, 5, which is symmetric around the lowest value.

Plotting the points

Once your table is complete, turn each column into a coordinate:

x = -2, y = 5 gives $(- 2, 5)$
x = -1, y = 0 gives $(- 1, 0)$
x = 0, y = -3 gives $(0, - 3)$
x = 1, y = -4 gives $(1, - 4)$
x = 2, y = -3 gives $(2, - 3)$
x = 3, y = 0 gives $(3, 0)$
x = 4, y = 5 gives $(4, 5)$

Example

Using a table to list coordinates

Use the completed table to write down the coordinates for the graph of $y=x^2-2x-3$ .

Match each x-value with the y-value directly underneath it.
Write each pair in the form $(x, y)$ .
The coordinates are:
$(4,5)(-2,5),\ (-1,0),\ (0,-3),\ (1,-4),\ (2,-3),\ (3,0),\ (4,5)$
Plot each point carefully on the coordinate grid.

Common Mistake

Joining points too early

Do not draw the curve until all the points have been plotted. One wrong point can make the whole curve look distorted.

Drawing the smooth curve

After plotting the points, draw a smooth curve through them. For $y=x^2-2x-3$ , the coefficient of $x^2$ is positive, so the parabola opens upwards.

The graph below shows the completed curve for this example, including the intercepts and turning point.

A labelled parabola for y equals x squared minus 2x minus 3, showing plotted points, x-intercepts, y-intercept and turning point.

Definition

Turning point and intercepts

The turning point is the point where the parabola changes direction. An x-intercept is where the graph crosses the x-axis. A y-intercept is where the graph crosses the y-axis.

Example

Drawing the graph from the points

Draw the graph of $y=x^2-2x-3$ using the completed table.

Draw a pair of axes with a suitable scale. The x-values go from -2 to 4, and the y-values go from -4 to 5, so make sure your grid includes these values.
Plot the coordinates:
$(4,5)(-2,5),\ (-1,0),\ (0,-3),\ (1,-4),\ (2,-3),\ (3,0),\ (4,5)$
Check that the points form a smooth U-shape rather than a straight line.
Draw a smooth curve through the points, making the bottom of the curve near $(1, - 4)$ .
Extend the curve slightly beyond the end points if the grid allows, but do not invent a completely different shape.

Tip

Pencil first

In an exam, draw the curve lightly in pencil first. If one point looks out of place, recheck its substitution before committing to the curve.

How to check your quadratic graph

There are three useful checks.

1. Check the y-intercept

The y-intercept happens when $x = 0$ . For $y=x^2-2x-3$ , substituting $x = 0$ gives $y = - 3$ , so the graph should cross the y-axis at $(0, - 3)$ .

2. Check the x-intercepts

The x-intercepts happen when $y = 0$ . From the table, $y = 0$ when x is -1 and 3, so the graph should cross the x-axis at $(- 1, 0)$ and $(3, 0)$ .

3. Check the turning point

The turning point should sit in the middle of the symmetry. Here the matching y-values are:

5 and 5
0 and 0
-3 and -3

So the lowest point is halfway between the two sides, at about $(1, - 4)$ .

Example

Spotting an error in a plotted point

A student draws $y=x^2-2x-3$ but plots the point for $x = 2$ at $(2, 3)$ . Explain why this is likely to be wrong.

Substitute $x = 2$ into the equation:
$y=2^2-2(2)-3$
Simplify the value:
$y = 4 - 4 - 3 = - 3$
The correct point is $(2, - 3)$ , not $(2, 3)$ .
Also, the table has symmetry: the points $(0, - 3)$ and $(2, - 3)$ should have the same y-value.

Common Mistake

Do not force a straight line

Even if the plotted points look nearly straight over a small section, a quadratic graph must be drawn as a smooth curve, not as joined straight-line segments.

What if the parabola opens downwards?

If the coefficient of $x^2$ is negative, the parabola opens downwards instead of upwards. For example, $y=-x^2+4$ has an upside-down U-shape.

Example

A downward quadratic

Complete a few values for $y=-x^2+4$ .

Substitute $x = - 2$ :
$y=-(-2)^2+4=-4+4=0$
Substitute $x = 0$ :
$y=-(0)^2+4=4$
Substitute $x = 2$ :
$y=-(2)^2+4=0$
The points $(- 2, 0)$ , $(0, 4)$ and $(2, 0)$ show that the graph has a highest point instead of a lowest point.

Exam technique

In the exam

Complete the table carefully, showing at least one substitution if working marks are available.
Plot every coordinate with a small, neat cross, then check for a smooth quadratic pattern.
Draw a single smooth curve through the points; do not use a ruler to join them with straight lines.

Self review

Check yourself

If $y=x^2+3x-4$ , what value of $y$ do you get when $x = - 1$ ?
Why should a quadratic graph be drawn as a curve rather than straight line segments?
How can the y-intercept help you check whether your graph is sensible?

Drawing Quadratic Graphs

What you'll learn

The basic idea

Prerequisite: coordinates

Substituting into a quadratic

Completing a table of values

Plotting the points

Drawing the smooth curve

How to check your quadratic graph

1. Check the y-intercept

2. Check the x-intercepts

3. Check the turning point

What if the parabola opens downwards?

Recap questions

You've reached the end

Drawing Quadratic Graphs

What you'll learn

The basic idea

Prerequisite: coordinates

Substituting into a quadratic

Completing a table of values

Plotting the points

Drawing the smooth curve

How to check your quadratic graph

1. Check the y-intercept

2. Check the x-intercepts

3. Check the turning point

What if the parabola opens downwards?

Recap questions

You've reached the end

Drawing Quadratic Graphs

What you'll learn

The basic idea

Prerequisite: coordinates

Substituting into a quadratic

Completing a table of values

Plotting the points

Drawing the smooth curve

How to check your quadratic graph

1. Check the y-intercept

2. Check the x-intercepts

3. Check the turning point

What if the parabola opens downwards?

Recap questions

You've reached the end

Drawing Quadratic Graphs Revision Guide

Drawing Quadratic Graphs

What you'll learn

The basic idea

Prerequisite: coordinates

Substituting into a quadratic

Completing a table of values

Plotting the points

Drawing the smooth curve

How to check your quadratic graph

1. Check the y-intercept

2. Check the x-intercepts

3. Check the turning point

What if the parabola opens downwards?

Recap questions

You've reached the end

Drawing Quadratic Graphs Revision Guide