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Investigate a quadratic: graph and features

Build a value table for a quadratic, plot a smooth curve and read its roots, intercept, turning point, symmetry, range and domain. A worked example links standard, factorised and completed-square forms before pupils investigate a second function.

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Follow the worked quadratic from its plotted points to its roots, intercept and turning point.

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Investigate a quadratic: graph and features

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Page 1

NAME
CLASS
DATE
LEARNING GOAL
I can draw the graph of a quadratic function
and use it to describe the function's key
features.
SUCCESS CRITERIA
I can square negatives correctly in a table of values.
I can plot points and join them with one smooth curve.
I can read roots, intercepts, turning point and range.
I can link algebraic forms to features of the graph.
1
Worked example: build the table for y = x2 βˆ’ 2x βˆ’ 3
Work out each row, then add down the column to get y. A negative squared is always positive.
x
βˆ’2
βˆ’1
0
1
2
3
4
x2
4
1
0
1
4
9
16
βˆ’2x
4
2
0
βˆ’2
βˆ’4
βˆ’6
βˆ’8
βˆ’3
βˆ’3
βˆ’3
βˆ’3
βˆ’3
βˆ’3
βˆ’3
βˆ’3
y
5
0
βˆ’3
βˆ’4
βˆ’3
0
5
2
Plot the points, then read the features
A
A
B
C
D
x
y
βˆ’2
βˆ’2 βˆ’1
βˆ’1
1
2
3
4
βˆ’4
βˆ’4
βˆ’3
βˆ’3
βˆ’2
βˆ’2
βˆ’1
βˆ’1
1
2
3
4
5
0
1 square = 1 unit on both axes. Plot each
point as a small cross, then draw one
smooth curve through them all: rounded at
the bottom, never pointed or flat.
A
ROOTS
x = βˆ’1 and x = 3
The curve meets the x-axis.
B
Y-INTERCEPT
(0, βˆ’3)
At x = 0, y is the constant.
C
TURNING POINT
(1, βˆ’4), minimum
The lowest point of the curve.
D
LINE OF SYMMETRY
x = 1
x = βˆ’2 and x = 4 both give 5.
R
RANGE
y β‰₯ βˆ’4
Nothing lies below it.
E
DOMAIN, ASYMPTOTES
All real x; none
It never levels off.
ONE FUNCTION, THREE FORMS
x2 βˆ’ 2x βˆ’ 3 = (x + 1)(x βˆ’ 3) = (x βˆ’ 1)2 βˆ’ 4
Factorised shows the roots: a bracket is
0 when x = βˆ’1 or x = 3.
Completed square shows the turning
point: (x βˆ’ 1)2 is smallest, 0, at x = 1.
Function graph investigation Β· Worked example
MATHEMATICS
AGES 14 TO 16
QUADRATIC GRAPHS
Function graph investigation
Plot a quadratic, then read its story from the graph.
PageΒ 1Β ofΒ 2

Page 2

1
Complete the table of values
Fill each row, then add down the columns, as on page 1.
4 marks
x
βˆ’5
βˆ’4
βˆ’3
βˆ’2
βˆ’1
0
1
2
3
x2
+2x
βˆ’8
βˆ’8
βˆ’8
βˆ’8
βˆ’8
βˆ’8
βˆ’8
βˆ’8
βˆ’8
βˆ’8
y
2
Draw the graph
Plot, join, label A to D.
3 marks
x
y
βˆ’5
βˆ’5 βˆ’4
βˆ’4 βˆ’3
βˆ’3 βˆ’2
βˆ’2 βˆ’1
βˆ’1
1
2
3
βˆ’9
βˆ’9
βˆ’8
βˆ’8
βˆ’7
βˆ’7
βˆ’6
βˆ’6
βˆ’5
βˆ’5
βˆ’4
βˆ’4
βˆ’3
βˆ’3
βˆ’2
βˆ’2
βˆ’1
βˆ’1
1
2
3
4
5
6
7
3
Read the features
7 marks
A
ROOTS
x = 
 and x = 
B
Y-INTERCEPT
( 0 , 
 )
C
TURNING
POINT
( 
 , 
 ) Β 
minimum 
maximum
D
LINE OF
SYMMETRY
x = 
 Β Check: ( 
 + 
 ) Γ· 2 = 
R
RANGE
E
DOMAIN,
ASYMPTOTES
Domain: 
 Β Asymptotes? 
yes 
no
Explain: why is the y-intercept always the constant term of a quadratic?
4
Connect the forms
4 marks
x2 + 2x βˆ’ 8 = (x + 
)(x βˆ’ 
)
This form shows the
x2 + 2x βˆ’ 8 = (x + 
)2 βˆ’ 
This form shows the
Check by expanding: do you get x2 + 2x βˆ’ 8 back?
GO FURTHER
What changes for y = βˆ’x2 + 4? Plot it on your grid
for βˆ’3 ≀ x ≀ 3 with a dashed curve. Name two
features that change and one that stays the same.
Function graph investigation Β· Total 18 marks
FUNCTION GRAPH INVESTIGATION
YOUR TURN
Investigate y = x2 + 2x βˆ’ 8
PageΒ 2Β ofΒ 2

A teaching sequence

Calculate and graph a quadratic function and connect its algebraic forms to key graph features.

Follow the worked quadratic from its plotted points to its roots, intercept and turning point.

Read the complete material

Build and plot the table

Calculate each table value for y = xΒ² βˆ’ 2x βˆ’ 3, remembering that a negative value squared is positive. Plot the coordinates and draw a smooth curve.

For a negative x, have you squared the whole value before subtracting 2x?

Follow the worked quadratic from its plotted points to its roots, intercept and turning point.

Read the complete material

Connect graph and forms

Read the roots, y-intercept, turning point and line of symmetry. Compare standard, factorised and completed-square forms to check the features.

Where do the factorised and completed-square forms show the roots and turning point?

Investigate the second curve

Complete and plot y = xΒ² + 2x βˆ’ 8, using its factorised and completed-square forms to check your answers.

Does expanding your factorised form recover xΒ² + 2x βˆ’ 8?

Compare an inverted curve

Try y = βˆ’xΒ² + 4 on the same grid and describe two changed features and one that stays. Use completed-square form to identify the turning point and verify that point on the graph.

Which feature changes for y = βˆ’xΒ² + 4, and which remains?

Before you teach

Look for the thinking behind an answer, choose the support your class needs and leave room for an independent response.

What to look for

Check table values, coordinate placement, curve shape and labels. Roots occur where the graph crosses the x-axis; the y-intercept occurs at x = 0. Confirm that the line of symmetry passes through the turning point and that the stated range reflects the minimum or maximum.

Give a little support

Use graph paper or a digital plotting tool to practise coordinates before completing the worksheet. Separate the calculations for xΒ², the linear term and the constant in different colours. Provide a table of square numbers if needed, then return to the complete equation.

Take the idea further

Ask pupils to create a quadratic with a chosen turning point or pair of roots, then write it in two forms and sketch the graph. They can compare how a negative leading coefficient changes the shape and range.

Things to keep in mind

The grids show a finite set of values and the graph is a representation of a continuous function. Read domain and range from the function, not only the visible grid. The prompts cover quadratic examples and do not replace broader function notation or proof.

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Another starting point

Explore a different teaching task from this selection.

Ideas to try with your class

Sample teaching perspectives on what to try, what to adapt and what to watch for.

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Bright Margins

The worked quadratic table leads to roots, y-intercept, turning point, symmetry and range, then shows the same function in standard, factorised and completed-square forms. The β€œnegative squared is positive” note targets a key calculation step.

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Chalk Dust & Checklists

For ages 14 to 16, the task connects values and graph shape to algebraic features: brackets show roots, while completed square shows the minimum. Pupils then use a second function to repeat the investigation.

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Glue Stick Club

Plot each table pair as a cross and draw one smooth curve, as instructed. Reading one feature from its form before checking the graph can help pupils see why the three expressions describe the same quadratic.

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Good Question

A plotted graph gives a visual check but should not replace algebraic reasoning or precise coordinates. Check negative substitutions and the stated domain/range interpretation before using the graph as evidence of a wider function understanding.

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