Types of Graphs (Edexcel GCSE Maths: Foundation)

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Naomi C

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Maths

Types of Graphs

What graphs do I need to know?

You need to be able to recognise the following lines:
- Straight lines
  - y = mx + c
  - Such as y = 3x + 2, y = 5x - 1, ...
  - Two important ones are y = x and y = -x
- Horizontal lines
  - y = c
  - Such as y = 4, y = -10, ...
- Vertical lines
  - x = k
  - Such as x = 2, x = -1, ...
You need to be able to recognise quadratic graphs
- y = x²
- y = -x²
- y = ax² + bx + c
You need to be able to recognise simple cubic graphs
- y = x³
- y = -x³
- y = ax³ + bx² + x + c
You also need to be able to recognise reciprocal graphs
- $y = \frac{1}{x}$ , where $x \neq 0$

Example of graphs including linear, quadratic, cubic and reciprocal

What does a quadratic graph look like?

The equation of a quadratic graph is y = ax² + bx + c
A quadratic graph has either a u-shape or an n-shape
- This type of shape is called a parabola
u-shapes are called positive quadratics
- because the number in front of x² is positive
  - For example, y = 2x² + 3x + 4
n-shapes are called negative quadratics
- because the number in front of x² is negative
  - For example, y = -3x² + 2x + 4
You can plot quadratic graphs using a table of values

Positive and negative quadratic graphs

What does a cubic graph look like?

The equation of a cubic graph is y = ax³ + bx² + cx + d
A cubic graph can have two points where it changes direction (turning points)
A positive cubic goes uphill (from the bottom left to the top right)
- The number in front of x³ is positive
  - For example, y = x³ - 3x² + 2x + 1
A negative cubic goes downhill (from the top left to the bottom right)
- The number in front of x³ is negative
  - For example, y = -x³ + 2x² - x + 5
You can plot cubic graphs using a table of values

What does a reciprocal graph look like?

The equation of the basic reciprocal graph is
- You cannot substitute in x = 0 (division by zero is not allowed)
  - $x \neq 0$
- You should not include x = 0 in a table of values
The shape of is shown below
- It has two two curved branches
  - The branches are L-shaped
- The branches never connect!

Worked example

In each of the cases below, state the letter of the graph that corresponds to the equation given.

(a)

y = x + 5

This is a straight-line graph, y = mx + c
The graph is a straight line going uphill and crosses the x-axis above (0,0)

Graph D

(b)

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

This is a quadratic graph, y = ax² + bx + c (a = -1, b = 3, c = 2)

The number in front of x² is negative so it has an n-shape

Graph C

(c)

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y = \frac{1}{x}

This is the reciprocal graph,

y = \frac{1}{x}

It has two L-shaped branches and no y-value when x = 0

Graph B

Quadratic Graphs

What are the key features of a quadratic graph?

The point where the graph turns is called the vertex
- Positive quadratics have a minimum point
  - The bottom of the u-shape
- Negative quadratics have a maximum point
  - The top of the n-shape
Quadratic graphs always have a vertical line of symmetry down the middle
- The equation of the vertical line of symmetry is x = k
  - k is the x-coordinate of the minimum or maximum point
Quadratic graphs do not have to cross the x-axis
- If they do, two x-intercepts are created, called roots
  - If the curve just touches the x-axis, only 1 root is created
- Roots are symmetric about the vertical line of symmetry
Quadratic graphs always have one y-intercept

Worked example

The graph of the equation $y = x^{2} - 5 x + 6$ is shown below.

Graph of quadratic

(a)

Write down the coordinates of the roots of the equation.

The roots of the equation are the x-intercepts of the graph

The graph crosses at x = 2 and x = 3

The roots of the graph are at (0, 2) and (0, 3)

(b)

Write down the equation of the line of symmetry.

The line of symmetry is a vertical line that occurs halfway between the x-intercepts

Find the x-value that is halfway between the roots

The x-intercepts are x = 2 and x = 3
Halfway between is x = 2.5

Write down the equation of the line of symmetry

x = 2.5

Drawing Graphs Using a Table

How do I draw a graph using a table of values?

To create a table of values
- Substitute different x-values into the equation
- This gives the y-values
To plot the points
- Use the x and y-values to mark crosses on the grid at the coordinates (x , y )
Draw a single smooth freehand curve
- Go through all the plotted points
- Make it the shape you would expect
  - For example, quadratic curves have a vertical line of symmetry
- Do not use a ruler for curves!

Which numbers should I be careful with?

For quadratic and cubic graphs, be careful substituting in negative numbers
Always put brackets around them and use BIDMAS
- For example, x = -3 in y = -x² + 8x
  - becomes y = -(-3)² + 8(-3)
  - which simplifies to -9 - 24
  - so y = - 33
For reciprocal graphs like , do not include x = 0
- You cannot divide by zero
  - You get an error on your calculator
- There is no value at x = 0
  - The L-shaped branches can't cross the y-axis
- An example is given below with $y = \frac{1}{x}$

$x$	-3	-2	-1	0	1	2	3
$y$	$- \frac{1}{3}$	$- \frac{1}{2}$	$- 1$	No value	$1$	$\frac{1}{2}$	$\frac{1}{3}$

How do I use the table function on my calculator?

Calculators can create tables of values for you
Find the table function
- Type in the graph equation (called the function, f(x))
  - Use the alpha button then X or x
  - Press = when finished
- If you are asked for another function, g(x), press = to ignore it
Enter the start value
- The first x-value in the table
- Press =
Enter the end value
- The last x-value in the table
Enter the step size
- How big the steps (gaps) are from one x-value to the next
- Press =
Then scroll up and down to see all the y-values

Exam Tip

If you find a point that doesn't seem to fit the shape of the curve, check your working!
If any y-values are given in the question, check that your calculations agrees with them.

Worked example

(a)

Complete the table of values for the graph of

y = 10 - 8 x^{2}

$x$	$- 1.5$	$- 1$	$- 0.5$	$0$	$0.5$	$1$	$1.5$
$y$		$2$					$- 8$

Use the table function on your calculator for

f (x) = 10 - 8 x^{2}

Start at -1.5, end at 1.5 and use steps of 0.5
On a non-calculator paper, substitute the x-values into the equation, for example x = -1.5

\begin{array}{rcl} y & = & 10 - 8 {(- 1.5)}^{2} \\ = & 10 - 8 \times 2.25 \\ = & 10 - 18 \\ = & - 8 \end{array}

$x$	$- 1.5$	$- 1$	$- 0.5$	$0$	$0.5$	$1$	$1.5$
$y$	-8	$2$	8	10	8	2	$- 8$

(b)

Plot the graph of

y = 10 - 8 x^{2}

on the axes below, for values of

x

from

- 1.5

1.5

Carefully plot the points from your table on to the grid
Note the different scales on the axes

Join the points with a smooth curve (do not use a ruler)

cie-igcse-2018-may-jun-1-7

(c)

Write down the equation of the line of symmetry of the curve.

There is a vertical line of symmetry about the y-axis

The equation of the y-axis is x = 0

x = 0

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Types of Graphs (Edexcel GCSE Maths: Foundation)

Revision Note

What graphs do I need to know?

What does a quadratic graph look like?

What does a cubic graph look like?

What does a reciprocal graph look like?

What are the key features of a quadratic graph?

How do I draw a graph using a table of values?

Which numbers should I be careful with?

How do I use the table function on my calculator?

You've read 0 of your 0 free revision notes

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Join the 100,000+ Students that ❤️ Save My Exams

Author: Naomi C