Quadratic graphs - KS3 Maths

Key points

An image of the graph y equals x squared plotted on a set of axes. — Image caption,
Quadratic graph lines are U- or Ո-shaped, which is called a parabola.

A graph is a visual representation of a quadratic equation in the form \(y = ax² + bx + c\) where the \(a\) and \(b\) are , and \(c\) is a .
A linear graph can be drawn using only three points. A quadratic graph is a curve, so more points are plotted to support accuracy.
The shape of a quadratic graph is called a , which looks like a U-shape. When the \(x²\) term is positive the graph is U-shaped. When \(x²\) is negative the graph is Ո-shaped. The graph has a vertical line of symmetry.
Understanding how to substitute values into an expression is an essential skill when using graphs to estimate values of \(x\) and \(y\).

How to draw the graphs 𝒚 = 𝒙² and 𝒚 = -𝒙²

There are no straight-line segments on a quadratic graph. The coordinates \((x, y)\) are plotted and joined by drawing a freehand curve.

Two consecutive horizontal points are joined by a rounded curve.
A single point is rounded off to avoid a pointed turn.

To draw the graph \(y = x²\):

Set up a table of values for \(x\) and \(y\), using the given values of \(x\)
Find the values for \(y\)
- For \(y = x²\), work out \(x²\) (which is \(x\) multiplied by itself).
- For \(y = -x²\), work out \(-x²\) (which is the negative value of \(x²\)).
Draw and label the axes to include the minimum and maximum values of \(x\) and \(y\).
The coordinates \((x, y)\) are read from the table and plotted on the axes.
Join the points, drawing a freehand curve through the points.
Label the graph.

Examples

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Image caption,
Draw the graph of the quadratic equation 𝒚 = 𝒙² for values of 𝒙 from -3 to 3. This is shown by the inequality -3 ≤ 𝒙 ≤ 3
Image caption,
Set up a table of values using the given values of 𝒙 from -3 to 3. There are two rows, one for the 𝒙 values and one for the 𝒚 values, which are 𝒙²
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Substitute each value of 𝒙 to find the 𝒚 values using the equation 𝒚 = 𝒙². This means that 𝒙 is multiplied by itself. (-3)² = 9, (-2)² = 4, (-1)² = 1, 0² = 0, 1² = 1, 2² = 4 and 3² = 9. The values in the table are all positive, because square numbers are always positive, and they mirror each other. Eg, (-3)² is the same as 3², and (-2)² is the same as 2²
Image caption,
Draw and label the axes to include the minimum and maximum values of 𝒙 and 𝒚. The 𝒙-axis is drawn using the given values from -3 to 3. The 𝒚-axis is drawn using values from 0 to 9, which are worked out using 𝒚 = 𝒙²
Image caption,
Use the pairs of values in the table to list the coordinates of the points to be plotted. The coordinates are (-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4) and (3, 9). Plot the coordinates on the graph.
Image caption,
Join the points, drawing a freehand curve through the points, and label the graph.
Image caption,
Draw the graph of the quadratic equation 𝒚 = -𝒙² for values of 𝒙 from -3 to 3. This is shown by the inequality -3 ≤ 𝒙 ≤ 3
Image caption,
Set up a table of values using the given values of 𝒙 from -3 to 3. There are two rows, one for the 𝒙 values and one for the 𝒚 values (-𝒙²). Substitute each value of 𝒙 to find the 𝒚 values, using the equation 𝒚 = -𝒙². As all square numbers (𝒙²) are positive, all the values of -𝒙² are negative. The values also mirror each other, -9, -4, -1, 0, -1, -4, -9...
Image caption,
Use the pairs of values in the table to list the coordinates of the points to be plotted. The coordinates are (-3, -9), (-2, -4), (-1, -1), (0, 0), (1, -1), (2, -4) and (3, -9). Plot the coordinates on the graph and join using a freehand curve. The parabola is Ո-shaped when 𝒚 = -𝒙²
Image caption,
There are no straight-line segments on a quadratic graph. Two consecutive horizontal points are joined by a rounded curve. A single point is rounded off to avoid a pointed turn.

Question

Which graph, A, B or C, shows \(y = x²\)?

The image shows three curved graphs, A, B and C. Graph A shows a set of axes. The horizontal axis is labelled x. The values go up in ones from negative three to positive three. The vertical axis is labelled y. The values go up in ones from zero to six. The curve passes through negative two comma six, negative one comma three, zero comma two, one comma three and two comma six. Graph B shows a set of axes. The horizontal axis is labelled x. The values go up in ones from negative three to positive three. The vertical axis is labelled y. The values go up in ones from negative two to four. The curve passes through negative two comma two, negative one comma negative one, zero comma negative two, one comma negative one and two comma two. Graph C shows a set of axes. The horizontal axis is labelled x. The values go up in ones from negative three to positive three. The vertical axis is labelled y. The values go up in ones from zero to six. The curve passes through negative two comma four, negative one comma one, zero comma zero, one comma one and two comma four.

How to draw a quadratic graph from a table of values

A quadratic equation in the form \(ax² + bx + c\) combines several .

A table of values is used to find the points to plot for a graph. A row is used for each term in the equation to work out the coordinates.

To draw a quadratic graph from an equation:

Set up a table of values using the given values of \(x\)
Use additional rows in the table to work out each term in the quadratic equation.
Substitute the values of \(x\) to evaluate each term, a row at a time.
Add up the terms of the quadratic expression, a column at a time, to find the value of \(y\)
Draw and label the axes to include the minimum and maximum values of \(x\) and \(y\)
Read the coordinates (\(x, y\)) from the table and plot on the axes.
Join the points, drawing a freehand curve through the points.
Label the graph.

Example

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Draw the graph of the quadratic equation 𝒚 = 𝒙² + 2𝒙 – 3 for values of 𝒙 from -4 to 2. This is shown by the inequality -4 ≤ 𝒙 ≤ 2
Image caption,
Set up a table of values using the given values of 𝒙 from -4 to 2. Each row of the table is a term in the quadratic equation. There is a row for the values of 𝒙², a row for the values of +2𝒙 and a row for – 3. The final row is for the values of 𝒚, which is the sum of the values of 𝒙², +2𝒙 and – 3
Image caption,
Substitute the values of 𝒙 to evaluate each term, one row at a time. The first row is the term 𝒙². This means that 𝒙 is multiplied by itself. (-4)² = 16, (-3)² = 9, (-2)² = 4, (-1)² = 1, 0² = 0, 1² = 1 and 2² = 4
Image caption,
The second row is for the term +2𝒙. This is 2 multiplied by 𝒙. 2 × -4 = -8, 2 × -3 = -6, 2 × -2 = -4, 2 × -1 = -2, 2 × 0 = 0, 2 × 1 = 2 and 2 × 2 = 4
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The third row is the term – 3. The term – 3 is a constant of 3 being subtracted. – 3 is written across the row.
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Add up the terms of the quadratic expression, a column at a time, to find the value of 𝒚. The value of 𝒚 when 𝒙 = -4 is found by adding the values of 𝒙² (16), +2𝒙 (-8) and – 3. The calculation is 16 – 8 – 3 = 5. Repeat the process for each given 𝒙-coordinate. The values of 𝒚 are 5, 0, -3, -4, -3, 0 and 5. Use the pairs of values in the table to list the (𝒙, 𝒚) coordinates of the points to be plotted. The coordinates are (-4, 5), (-3, 0), (-2, -3), (-1, -4), (0, -3), (1, 0) and (2, 5).
Image caption,
Draw and label the axes to include the minimum and maximum values of 𝒙 and 𝒚. The 𝒙-axis is drawn using the given values from -4 to 2. The 𝒚-axis is drawn using values from -4 to 5, worked out using 𝒙² +2𝒙 – 3
Image caption,
Plot the coordinates (-4, 5), (-3, 0), (-2, -3), (-1, -4), (0, -3), (1, 0) and (2, 5).
Image caption,
Join the points, drawing a freehand curve through the points. Label the graph.

Questions

Question 1:

Complete the table for \(y = x² – 4x + 5\) for values of \(x\) from –1 to 3

y equals x squared subtract four x plus five. Negative one is less than or equal to x, less than or equal to three. Written below: a table with five rows. The first row is labelled x, and has columns from negative one to positive three. The second row is labelled x squared. The first cell is empty, the rest of the cells have been completed, zero, one, four, nine. The third row is labelled subtract four x. The fourth row is labelled plus five and all cells have been populated with plus five. The fifth row is labelled y. The remaining cells are empty. In the function the x squared is coloured pink, the subtract four x is green and the plus five is purple.

Question 2:

Complete the table for \(y = 2x² + 3x – 4\) for values of \(x\) from –2 to 2

y equals two x squared plus three x subtract four. Negative two is less than or equal to x, less than or equal to two. Written below: a table with five rows. The first row is labelled x, and has columns from negative two to positive two. The second row is labelled two x squared. The third row is labelled plus three x. The cells have been completed, negative six, negative three, zero, positive three, positive six. The fourth row is labelled negative four and all cells have been populated with negative four. The fifth row is labelled y. The remaining cells are empty. In the function the two x squared is coloured pink, the plus three x is green and the subtract four is purple.

Use a quadratic graph to estimate values of 𝒙 and 𝒚

A quadratic graph can be used to find an estimate of the value of \(x\) from a given value of \(y\), or the value of \(y\) from a given value of \(x\)

A value of \(x\) will give one \(y\)-value.

A value of \(y\) will give two possible values of \(x\) because a quadratic graph is symmetrical.

To find a \(y\)-value from a given \(x\)-value:
1. On the \(x\)-axis, locate the given amount.
2. Draw a vertical line, using a ruler, from the given amount to the curve.
3. Draw a horizontal line, using a ruler, from the curve across to the \(y\)-axis.
4. Read the value on the \(y\)-axis.
To find an \(x\)-value from a given \(y\)-value:
1. On the \(y\)-axis, locate the given amount.
2. Draw a horizontal line, using a ruler, from the given amount across to the curve. It will cross the curve in two places.
3. Draw a vertical line, using a ruler, from the curve down to the \(x\)-axis.
4. Read the values on the \(x\)-axis.

Examples

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Use the graph of 𝒚 = 2𝒙² – 7𝒙 + 3 to find the value of 𝒚 when 𝒙 = 2
Image caption,
On the 𝒙-axis, locate the given value of 𝒙 (2). Draw a vertical line, using a ruler, down from 2 on the 𝒙-axis to the curve.
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Draw a horizontal line, using a ruler, from the curve across to the 𝒚-axis. Read the value on the 𝒚-axis. When 𝒙 is 2, 𝒚 is -3
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Use the graph of 𝒚 = 𝒙² – 3𝒙 – 2 to find the values of 𝒙 when 𝒚 = 2
Image caption,
On the 𝒚-axis, locate the given amount (2). Draw a horizontal line, using a ruler, from 2 on the 𝒚-axis across to the curve. It will cross the curve in two places.
Image caption,
Draw vertical lines, using a ruler, from the curve down to the 𝒙-axis. Read the values on the 𝒙-axis. When 𝒚 = 2, 𝒙 = -1 and 𝒙 = 4

Question

Use the graph of \(y = 3 + 4x – x²\) to find the values of \(x\) when \(y = 3\)

The image shows a quadratic graph drawn on a set of axes. The horizontal axis is labelled x. The values go up in ones from negative one to positive five. The vertical axis is labelled y. The values go up in ones from negative one to positive seven. The curve passes through zero comma three, one comma six, two comma seven, three comma six and four comma three. Written right: y equals three plus four x subtract x squared. Written left: y equals three, x equals question mark.

Practise reading and plotting quadratic graphs

Quiz

Practise reading and plotting quadratic graphs with this quiz. You may need a pen and paper to help you with your answers.

Real-life maths

An image of a basketball at different points in its motion having been thrown towards a hoop. The shape it makes is a parabola. — Image caption,
A basketball will create a parabola in the air when thrown into a hoop.

A parabola is the shape of a quadratic graph.

Understanding this shape can be helpful to many types of sports athletes. An object thrown or kicked through the air will create a parabola.

A netball or basketball player could create digital quadratic graphs to analyse their shooting technique, for example. A rugby or football player could do the same when practising conversions or free kicks.

Doing this can help sports players adjust their throws or kicks and improve their accuracy.