Lesson 16.6 · 60 min

Sketching Quadratic Graphs

Sketch any parabola without a table of values: find where it crosses the axes, decide its shape, and complete the square to find the turning point — then check it sits on the line of symmetry.

In this lesson you will learn to…

Find where a quadratic crosses the $x$ -axis (put $y = 0$ ) and the $y$ -axis (put $x = 0$ )
Decide whether the graph is ∪ or ∩ from the sign of the $x^{2}$ term
Complete the square, $y = (x + a)^{2} + b$ , and read off the turning point $(- a, b)$
Draw a sketch as a smooth, symmetrical curve with every key point labelled
Recognise a curve with no roots, using the discriminant or the completed square
Stretch: sketch ∩ curves, complete the square when the $x^{2}$ term is not 1, and find an equation from a sketch

The four steps

In class you sketched curves without a table of values. Four facts are enough to fix the whole curve:

Step	What to do	What it gives you
①	put $y = 0$ and solve	the roots — where it crosses the $x$ -axis
②	put $x = 0$	the $y$ -intercept — always just $c$
③	look at the sign of $x^2$	the shape: ∪ or ∩
④	complete the square	the turning point

Completing the square

Writing $y = x^2 + bx + c$ as $y = (x + a)^2 + b$ shows the turning point straight away. A square is never negative, so $(x + a)^2$ is smallest — zero — when $x = -a$ . Then $y = b$ . So the turning point is $(-a, b)$ .

Try it

Move the curve with the completed square

a, inside the bracket (x + a)²-2

b, added on the end-1

Shape: +1 is ∪, −1 is ∩1

Completed square: y = (x − 2)² − 1
Expanded: y = x² − 4x + 3
Turning point: (2, -1), a minimum
Line of symmetry: x = 2
y-intercept: (0, 3)
Roots: x = 1 and x = 3

This starts on $y = (x - 2)^{2} - 1$ , the class example. Slide $a$ and watch the turning point move the opposite way: $(x - 2)^{2}$ puts it at $x = + 2$ . Slide $b$ and it moves up and down by exactly $b$ . Then try to make the class curves $y = x^{2} + 2 x - 5$ and $y = x^{2} + 2 x - 24$ — what are $a$ and $b$ ?

To complete the square: halve the coefficient of $x$ to get the number in the bracket, then take away its square.

$x^2 + 6x + 1 = (x + 3)^2 - 9 + 1 = (x + 3)^2 - 8$

Putting it together

Drawing the sketch

A sketch does not need a ruler or a scale, but it does need to be a smooth, symmetrical curve with every key point labelled. A parabola is symmetrical about a vertical line through its turning point, so the turning point sits exactly halfway between the roots.

When there are no roots

Some parabolas never reach the $x$ -axis. You can tell in two ways: the discriminant $b^2 - 4ac$ is negative, or — for a ∪ curve — the completed square has a positive number on the end, so the lowest point is above the axis.

Upside-down curves

When the $x^2$ term is negative the curve is a ∩ with a maximum point. Take out $-1$ before you factorise, and use the halfway rule to find the turning point.

Stretch: exam-style extensions

When the $x^2$ term is not 1

Take the number out of the first two terms only, complete the square inside the bracket, then multiply back out.

Finding the equation from a sketch

If you know the roots $p$ and $q$ , the equation is $y = a(x - p)(x - q)$ . Use one more point — usually the $y$ -intercept — to find $a$ .

“Hence show that …”

Completing the square also proves facts. Because $(x + a)^2 \geq 0$ , the expression $(x + a)^2 + b$ can never be smaller than $b$ .

Key rules and formulas

The four steps (from class)
① Roots: put $y = 0$ and solve — factorise, or use the formula. ② $y$ -intercept: put $x = 0$ (it is just $c$ ). ③ Shape: $+ x^{2}$ gives ∪, $- x^{2}$ gives ∩. ④ Turning point: complete the square.
Completing the square
$x^{2} + b x + c = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2} + c$
Halve the $x$ coefficient for the bracket, then take its square away. In the form $y = (x + a)^{2} + b$ the turning point is $(- a, b)$ : flip the sign inside the bracket, keep the number outside.
Symmetry check
$x_{turning point} = \frac{root _{1} + root _{2}}{2}$
The turning point is always exactly halfway between the roots. Use it to check your completed square, and to place the bottom of the curve in your sketch.
How to draw the sketch
A smooth, symmetrical curve — never straight lines meeting at a point. Label the roots, the $y$ -intercept and the turning point with their coordinates. It does not need to be to scale.
No roots?
If $b^{2} - 4 a c < 0$ , or the completed square has a positive number on the end of a ∪ curve, the curve never meets the $x$ -axis. Sketch it using the $y$ -intercept and turning point.
Exact roots
If it will not factorise, the formula gives surds: for $x^{2} + 2 x - 5$ , $x = \frac{- 2 \pm 24}{2} = - 1 \pm 6$ . Give exact answers on a non-calculator paper; decimals ( $1.45$ and $- 3.45$ ) only if asked.

Try it

Completing the square explorer

a, inside the bracket (x + a)²-2

b, added on the end-1

Shape: +1 is ∪, −1 is ∩1

Completed square: y = (x − 2)² − 1
Expanded: y = x² − 4x + 3
Turning point: (2, -1), a minimum
Line of symmetry: x = 2
y-intercept: (0, 3)
Roots: x = 1 and x = 3

Slide $a$ and $b$ in $y = (x + a)^{2} + b$ . The red dot is the turning point $(- a, b)$ ; the dashed line is the line of symmetry. Switch the shape to −1 for a ∩ curve.

Worked examples

Worked example 1

The class example: $y = x^{2} - 4 x + 3$

Sketch

y = x^{2} - 4 x + 3

, showing where it crosses the axes and the turning point.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

When it will not factorise: $y = x^{2} + 2 x - 5$

Sketch

y = x^{2} + 2 x - 5

. Give the roots exactly.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

A ∩ curve: $y = - x^{2} - 2 x + 3$

Sketch

y = - x^{2} - 2 x + 3

Try it yourself first, then reveal the solution one step at a time.

Practice: sketching quadratic graphs

16 questions

Questions 1–11 are at class level; 12–16 are stretch questions at exam level. For every curve: roots (y = 0), y-intercept (x = 0), shape, then the turning point by completing the square — and check it is halfway between the roots.

Question 1 of 16Type the answerFoundation
Where does $y = x^{2} - 5 x + 6$ cross the $y$ -axis? Give the $y$ -value.
Your answer
Question 2 of 16Fill the gaps
Find the roots of $y = x^{2} - 5 x + 6$ .
$x =$ Gap a and $x =$ Gap b
Question 3 of 16Fill the gaps
Complete the square.
$x^{2} - 8 x + 10 = (x -$ Gap a $)^{2} -$ Gap b
Question 4 of 16Fill the gaps
Find the turning point of $y = x^{2} + 10 x + 21$ .
Turning point $=$ ( Gap a , Gap b )
Question 5 of 16True or false
The turning point of $y = (x - 4)^{2} + 1$ is $(- 4, 1)$ .
Question 6 of 16Multiple choice
Which describes the graph of $y = 3 + 2 x - x^{2}$ ?
Question 7 of 16Type the answer
A parabola crosses the $x$ -axis at $x = - 3$ and $x = 7$ . What is the $x$ -coordinate of its turning point?
Your answer
Question 8 of 16Explore and match
Move the sliders to make the curve $y = x^{2} + 6 x + 5$ . (Complete the square first.)
Move the controls until you match the question, then press Check.
a, inside the bracket (x + a)²0
b, added on the end0
Completed square
y = x²
Expanded
y = x²
Turning point
(0, 0), a minimum
Line of symmetry
x = 0
y-intercept
(0, 0)
Roots
one, at x = 0 (the curve touches the x-axis)
Question 9 of 16Explore and match
Build the sketch of $y = x^{2} - 2 x - 8$ : drag both roots R, the $y$ -intercept Y and the turning point T.
Move the controls until you match the question, then press Check.
Drag R along the x-axis to each root, Y up or down the y-axis, and T to the turning point.
Roots (R)
x = -1 and x = 2
y-intercept (Y)
(0, 2)
Turning point (T)
(3, -3)
Question 10 of 16Step by step
Sketch $y = x^{2} + 2 x - 3$ .
1. The roots are …
Question 11 of 16Multiple choice
$x^{2} - 4 x + 9 = (x - 2)^{2} + 5$ . How many times does $y = x^{2} - 4 x + 9$ cross the $x$ -axis?
Question 12 of 16Type the answerChallenge
Stretch. Find the minimum value of $x^{2} - 6 x + 14$ .
Your answer
Question 13 of 16Fill the gapsChallenge
Stretch. Complete the square.
$2 x^{2} + 12 x + 7 = 2 (x +$ Gap a $)^{2} -$ Gap b
Question 14 of 16Fill the gapsChallenge
Stretch. Find the turning point of $y = - x^{2} + 4 x + 5$ , and say whether it is a maximum or a minimum.
( Gap a , Gap b ), a Gap c
Question 15 of 16Step by stepChallenge
Stretch. A curve crosses the $x$ -axis at $x = 2$ and $x = 6$ , and the $y$ -axis at $(0, 24)$ . Find its equation.
1. Write $y = a (x - 2) (x - 6)$ and put $x = 0$ , $y = 24$ . What is $a$ ?
  Step 1 answer
Question 16 of 16Fill the gapsChallenge
Stretch. Find the exact roots of $y = x^{2} - 6 x + 4$ .
$x = 3 \pm$ Gap a

In this lesson you will learn to…

The four steps

Completing the square

Move the curve with the completed square

Putting it together

Drawing the sketch

When there are no roots

Upside-down curves

Stretch: exam-style extensions

When the x2x^2x2 term is not 1

Finding the equation from a sketch

“Hence show that …”

Key rules and formulas

The four steps (from class)

Completing the square

Symmetry check

How to draw the sketch

No roots?

Exact roots

Completing the square explorer

Worked examples

Practice: sketching quadratic graphs

When the $x^2$ term is not 1