19.1 How to Plot a Quadratic Graph Using a Table of Values

🟦 Concept

A quadratic graph can be drawn by calculating several coordinate points and plotting them on a coordinate grid.

For a quadratic equation:$$y=ax^2+bx+c$$

choose suitable values of $x$, calculate the corresponding values of $y$, and record the results in a table of values.

Then plot the coordinates and join them with a smooth curve.

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🟩 Example

Consider:$$y=x^2-4x+3$$

Choose these $x$-values:$$x=0,\ 1,\ 2,\ 3,\ 4$$

Now calculate $y$ for each value.

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1. When $x=0$

$$y=(0)^2-4(0)+3$$$$y=3$$

So the coordinate is:$$(0,3)$$

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2. When $x=1$

$$y=(1)^2-4(1)+3$$$$y=1-4+3$$$$y=0$$

So:$$(1,0)$$

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3. When $x=2$

$$y=(2)^2-4(2)+3$$$$y=4-8+3$$$$y=-1$$

So:$$(2,-1)$$

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4. When $x=3$

$$y=(3)^2-4(3)+3$$$$y=9-12+3$$$$y=0$$

So:$$(3,0)$$

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5. When $x=4$

$$y=(4)^2-4(4)+3$$$$y=16-16+3$$$$y=3$$

So:$$(4,3)$$

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🟪 Table of Values

$x$	0	1	2	3	4
$y$	3	0	−1	0	3

Therefore, the points to plot are:$$(0,3),\ (1,0),\ (2,-1),\ (3,0),\ (4,3)$$

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🟨 Plot the Points

Plot each coordinate accurately on the graph.

Notice that:$$(0,3)\quad\text{and}\quad(4,3)$$

have the same $y$-value.

Also:$$(1,0)\quad\text{and}\quad(3,0)$$

have the same $y$-value.

This happens because the quadratic graph is symmetrical.

The middle point is:$$(2,-1)$$

which is the turning point.

The line of symmetry is:$$x=2$$

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🟧 Draw the Curve

After plotting the points, join them using a smooth curved line.

Do not join the coordinates using straight line segments.

For this example, the finished graph opens upwards because:$$a=1>0$$

The key features are:$$\text{Turning point}=(2,-1)$$$$\text{Roots}=(1,0)\text{ and }(3,0)$$$$\text{y-intercept}=(0,3)$$$$\text{Line of symmetry}=x=2$$

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💡 Exam Tip

Choose enough $x$-values on both sides of the turning point.

This helps you draw the parabola accurately and makes its symmetry easier to see.

If the question already provides a table, carefully calculate any missing y-values before plotting the graph.

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⚠️ Common Mistake

Do not draw:

point → straight line → point → straight line

A quadratic graph is a smooth curve.

Also, make sure you:

• use the correct scale;
• plot each coordinate accurately;
• label both axes;
• draw a thin, smooth curve through the plotted points.

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⭐ Quick Summary

To plot a quadratic graph using a table of values:

Choose x-values

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Substitute each xx-value into the equation

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Calculate the corresponding y-values

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Write the coordinates in a table

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Plot the coordinates

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Join them with a smooth quadratic curve

For:$$y=x^2-4x+3$$

the table gives:$$(0,3),\ (1,0),\ (2,-1),\ (3,0),\ (4,3)$$

These points form a smooth upward-opening parabola.