19.2 How to Draw a Quadratic Graph?

A quadratic graph may look complicated at first, but it becomes much easier when you follow a clear method.

The basic process is:

Make a table → Calculate the values → Plot the points → Draw a smooth curve

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🔵 Concept

A quadratic graph is the graph of a quadratic function.

A quadratic function is usually written in the form:$$y=ax^2+bx+c$$

where:$$a\neq0$$

The graph of a quadratic function is called a parabola.

A parabola can open either upwards or downwards.

If:$$a>0$$

the parabola opens upwards.

If:$$a<0$$

the parabola opens downwards.

For example:$$y=x^2$$

opens upwards, while:$$y=-x^2$$

opens downwards.

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🟢 Key Idea

To draw a quadratic graph, we need to find several coordinates that lie on the curve.

Suppose the quadratic function is:$$y=x^2-4x+3$$

Choose a value of $x$, substitute it into the function, and calculate the corresponding value of $y$.

The two values form a coordinate:$$(x,y)$$

Repeat this process for several values of $x$.

Once you have enough coordinates, plot them on the graph and draw a smooth curve through them.

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🟣 Example 1: Drawing a Quadratic Graph

Draw the graph of:$$y=x^2-4x+3$$

for:$$-1\leq x\leq5$$

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🟨 Step 1: Make a Table of Values

We are given:$$-1\leq x\leq5$$

Therefore, we can use the integer values:$$x=-1,\ 0,\ 1,\ 2,\ 3,\ 4,\ 5$$

Start by making a table.

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

We calculate each missing value using:$$y=x^2-4x+3$$

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🟨 Step 2: Calculate the Values of $y$

When $x=-1$

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

So the coordinate is:$$(-1,8)$$

When $x=0$

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

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

When $x=1$

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

So the coordinate is:$$(1,0)$$

When $x=2$

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

So the coordinate is:$$(2,-1)$$

When $x=3$

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

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

When $x=4$

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

So the coordinate is:$$(4,3)$$

When $x=5$

$$y=(5)^2-4(5)+3$$$$y=25-20+3$$$$y=8$$

So the coordinate is:$$(5,8)$$

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🟨 Step 3: Complete the Table

The completed table is:

$x$	$-1$	$0$	$1$	$2$	$3$	$4$	$5$
$y$	$8$	$3$	$0$	$-1$	$0$	$3$	$8$

Therefore, the coordinates are:$$(-1,8),\quad(0,3),\quad(1,0),\quad(2,-1),\quad(3,0),\quad(4,3),\quad(5,8)$$

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🟨 Step 4: Choose a Suitable Scale

Before plotting the points, look at the values that need to fit on the graph.

For the $x$-axis:$$-1\leq x\leq5$$

For the $y$-axis:$$-1\leq y\leq8$$

For this example, a convenient scale could be:

x-axis: 1 large square = 1 unit
y-axis: 1 large square = 1 unit

Always check the scale carefully before plotting.

Do not assume that every square represents $1$ unit.

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🟨 Step 5: Plot the Coordinates

Plot each of the following points:$$(-1,8)$$$$(0,3)$$$$(1,0)$$$$(2,-1)$$$$(3,0)$$$$(4,3)$$$$(5,8)$$

Remember that a coordinate is written as:$$(x,y)$$

The $x$-coordinate tells you how far to move horizontally.

The $y$-coordinate tells you how far to move vertically.

For example, for:$$(2,-1)$$

move horizontally to:$$x=2$$

and then vertically down to:$$y=-1$$

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🟨 Step 6: Draw a Smooth Curve

Once all the points have been plotted, draw one smooth curve through them.

Do not join the points with separate straight lines.

A quadratic graph is a smooth parabola.

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

the coefficient of $x^2$ is positive:$$a=1$$

Therefore:$$a>0$$

so the parabola opens upwards.

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🟢 What Can We Read From the Graph?

Once the graph has been drawn, we can identify several important features.

1. Roots or x-Intercepts

The roots are where the graph meets the $x$-axis.

At the $x$-axis:$$y=0$$

For our graph, this happens when:$$x=1$$

and:$$x=3$$

Therefore, the roots are:$$x=1,\qquad x=3$$

These are also the solutions of:$$x^2-4x+3=0$$

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2. Turning Point

The lowest point of the graph is:$$(2,-1)$$

This is the turning point.

Since the parabola opens upwards, it is also the minimum point.

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3. Line of Symmetry

A quadratic graph is symmetrical about a vertical line through its turning point.

Since the turning point is:$$(2,-1)$$

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

The line of symmetry is also commonly called the axis of symmetry.

Notice that:$$y(0)=y(4)=3$$

and:$$y(1)=y(3)=0$$

The points on opposite sides of the line of symmetry have matching $y$-values.

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4. y-Intercept

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

At the $y$-axis:$$x=0$$

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

Therefore, the $y$-intercept is:$$(0,3)$$

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🟠 Exam Tip

In an exam, Cambridge may provide a table with some values already completed and ask you to calculate the missing values.

Complete the table before drawing the graph.

One incorrect value can cause a point to be plotted in the wrong place and affect the shape of the parabola.

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🟠 Exam Tip: Use Symmetry to Check Your Values

A quadratic graph is symmetrical.

For our example:$$y(0)=y(4)=3$$

and:$$y(1)=y(3)=0$$

If values that should match are different, check your calculations.

This is a quick way to detect an error.

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🟠 Exam Tip: Check the Scale

Never assume that every square represents $1$ unit.

For example, an axis may be labelled:$$0,\ 2,\ 4,\ 6,\ 8$$

or:$$0,\ 5,\ 10,\ 15,\ 20$$

Always check the scale on both axes before plotting the points.

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🔴 Common Mistake: Negative Numbers

Be careful when substituting negative values.

Suppose:$$x=-2$$

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

Substitute using brackets:$$y=(-2)^2+3(-2)-4$$

Since:$$(-2)^2=4$$

we have:$$y=4-6-4$$$$y=-6$$

Remember the important difference:$$(-2)^2=4$$

but:$$-2^2=-4$$

Using brackets around negative numbers helps prevent sign errors.

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🔴 Common Mistake: Joining the Points With Straight Lines

Do not use a ruler to join each plotted point to the next.

A quadratic graph should be a smooth continuous curve.

The points guide the shape of the parabola.

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🔴 Common Mistake: Drawing a Thick Curve

Avoid repeatedly drawing over the same line.

A thick or fuzzy curve makes it difficult to read accurate values from the graph.

This can cause problems when finding:

• roots,
• turning points,
• intersections,
• coordinates.

Use a sharp pencil and draw a thin, smooth curve.

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🧠 Easy Method to Remember

Remember these five steps:

TABLE → CALCULATE → PLOT → CHECK → CURVE

TABLE — Write the required $x$-values.

CALCULATE — Substitute each $x$-value to find $y$.

PLOT — Plot each coordinate $(x,y)$.

CHECK — Check the scale and your plotted points.

CURVE — Draw one smooth parabola.

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❓ Question & Answer

Q1. What is a quadratic graph called?

A quadratic graph is called a parabola.

Q2. What is the general form of a quadratic function?

$$y=ax^2+bx+c$$

where:$$a\neq0$$

Q3. How do I know whether the parabola opens upwards or downwards?

Look at the coefficient $a$.

If:$$a>0$$

the parabola opens upwards.

If:$$a<0$$

the parabola opens downwards.

Q4. Should I use a ruler to join the plotted points?

No. Draw a smooth curve through the points.

Q5. What is the turning point?

The turning point is where the parabola changes direction.

An upward-opening parabola has a minimum turning point, while a downward-opening parabola has a maximum turning point.

Q6. What are the roots of a quadratic graph?

The roots are the $x$-values where the graph meets the $x$-axis.

At the roots:$$y=0$$

Q7. Does every quadratic graph have two roots?

No. A quadratic graph may cross the $x$-axis twice, touch it once, or not meet it at all.

Q8. Why should I use brackets when substituting negative numbers?

Because:$$(-3)^2=9$$

but:$$-3^2=-9$$

Using brackets helps prevent sign errors.

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✅ Final Check

Before finishing your quadratic graph, check:

• Have I completed the table correctly?
• Have I substituted negative numbers using brackets?
• Have I checked the scale on both axes?
• Have I plotted every coordinate accurately?
• Have I used $x$ first and $y$ second?
• Have I drawn one smooth curve?
• Is my curve thin and clear?
• Have I avoided joining the points with straight lines?

If all of these are correct, your quadratic graph is complete.

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