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Negative numbers Substitution CoordinatesAddition and subtraction

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Here we will learn about **plotting graphs**, including how to plot straight-line graphs and the graphs of curves including quadratic graphs and cubic graphs

There are also plotting graphs worksheets and exam-style questions based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Plotting graphs** allows us to accurately plot coordinates onto a grid to produce the graph of a function.

We can plot a variety of different graphs. Linear functions which produce straight-line graphs but we can also plot other polynomials such as quadratic functions or cubic functions and exponential functions.

To do this we need to find x -coordinates and their y -coordinates.

These are plotted on x-y axes and the points are joined up.

For example,

y = 3x + 2

In order to plot a graph:

**Find the coordinates.****Plot the coordinates on the axes.****Join up the coordinates.**

Get your free plotting graphs worksheet of 20+ sketching surds questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free plotting graphs worksheet of 20+ sketching surds questions and answers. Includes reasoning and applied questions.

COMING SOONComplete the table of values for y=2x+3 and then on the grid draw the graph of y=2x+3:

**Find the coordinates.**

We need to find the missing y -coordinates. We do this by substituting the x -values into the equation.

For example,

When x = -1

When y = 2(-1) + 3 = -2 + 3 = 1

2**Plot the coordinates on the axes.**

The x -values and y -values in the table give coordinates to plot on the grid.

(-2,-1), \ (-1, 1) and so on. The points should be plotted with small crosses.

3**Join up the coordinates.**

The coordinates need to be joined up with a straight-line. The line can be extended beyond the first and last points.

On the grid draw the graph of y=3x-4 for values of x from -2 to 4:

**Find the coordinates.**

We need to make our own table. We can choose at least 3 \ x -values. Simple x -values such as 0, 1, and 2 will be sensible. We can then find the corresponding y -coordinates. We do this by substituting the x -values into the equation.

For example,

When x=1

y=3(1)-4=3-4=-1

**Plot the coordinates on the axes.**

The x -values and y -values in the table give coordinates to plot on the grid.

(0,-4), \ (1, -1) and (2,2). The points should be plotted with small crosses.

**Join up the coordinates.**

The coordinates need to be joined up with a straight-line. The line can be extended beyond the first and last points to cover the whole of the grid.

Complete the table of values for y=x^{2}-3x and then on the grid draw the graph of y=x^{2}-3x.

**Find the coordinates.**

We need to find the missing y -coordinates. We do this by substituting the x -values into the equation.

For example,

When x=-1

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

**Plot the coordinates on the axes.**

The x -values and y -values in the table give coordinates to plot on the grid.

(-1,4), \ (0, 0) and so on. The points should be plotted with small crosses.

**Join up the coordinates.**

The coordinates need to be joined up with a single smooth curve. The curve can be extended beyond the first and last points.

Complete the table of values for y=x^{3}+2x^{2}-3x and then on the grid draw the graph of y=x^{3}+2x^{2}-3x:

**Find the coordinates.**

We need to find the missing y -coordinates. We do this by substituting the x -values into the equation.

For example,

When x=-2

y=(-2)^3+2(-2)^2-3(-2)=-8+8+6=6

**Plot the coordinates on the axes.**

The x -values and y -values in the table give coordinates to plot on the grid.

(-3,0), \ (-2, 6) and so on. The points should be plotted with small crosses.

**Join up the coordinates.**

Complete the table of values for y=2^x and then on the grid draw the graph of y=2^x:

**Find the coordinates.**

For example,

When x=-1

y=2^{-1} = 0.5

**Plot the coordinates on the axes.**

The x -values and y -values in the table give coordinates to plot on the grid.

(-1,0.5), \ (0, 1) and so on. The points should be plotted with small crosses.

**Join up the coordinates.**

In order to plot a graph using the intercept method:

**Substitute**\bf{x=0}**to find the**\textbf{y}**-intercept.****Substitute**\bf{y=0}**to find the**\textbf{x}**-intercept.****Join the two points, then substitute the values of a third point to check your answer.**

On the grid draw the graph of 2x+3y=12.

**Substitute ** \bf{x=0} ** to find the ** \textbf{y} **-intercept.**

Because of the way the equation of the straight-line is written, it is simple to substitute x=0 to find where the line crosses the y -axis.

\begin{aligned} 2x+3y &= 12 \\\\ 2(0)+3y &=12\\\\ 3y&=12\\\\ y&=4 \end{aligned}

This means the line crosses the y -axis at 4. The y -intercept is 4.

**Substitute ** \bf{y=0} ** to find the ** \textbf{x} **-intercept.**

Similarly it is simple to substitute y=0, to find where the line crosses the x -axis.

\begin{aligned} 2x+3y &= 12 \\\\ 2x+3(0) &=12\\\\ 2x&=12\\\\ x&=6 \end{aligned}

This means the line crosses the x -axis at 6. The x -intercept is 6.

**Join the two points, then substitute the values of a third point to check your answer.**

Join the two points with a straight line.

Always check your answer by substituting the values of a third point into the original equation.

The point (3,2) lies on the line we have drawn. We substitute x=3, \ y=2 into the equation 2x+3y=12 to check.

\begin{aligned} 2x+3y &= 12 \\\\ 2(3)+3(2) &=12\\\\ 6+6&=12 \end{aligned}

The statement is correct, so we know that our line is drawn correctly.

Draw the graph of 2x+3y=18.

**Substitute ** \bf{x=0} ** to find the ** \textbf{y} **-intercept.**

Substitute x=0 into 2x+3y=18.

\begin{aligned} 2x+3y &= 18 \\\\
2(0)+3y &=18\\\\
3y&=18\\\\
y&=6
\end{aligned}

This means the line crosses the y -axis at 4. The y -intercept is 4.

**Substitute ** \bf{y=0} ** to find the ** \textbf{x} **-intercept.**

Substitute y=0 into 2x+3y=18.

\begin{aligned} 2x+3y &= 18 \\\\
2x+3(0) &=18\\\\
2x&=18\\\\
x&=9
\end{aligned}

This means the line crosses the x -axis at 9. The x -intercept is 9.

**Join the two points, then substitute the values of a third point to check your answer.**

Join the two points with a straight line:

Always check your answer by substituting the values of a third point into the original equation.

The point (3,4) lies on the line we have drawn. We substitute x=3, \ y=4 into the equation 2x+3y=18 to check.

\begin{aligned} 2x+3y &= 18 \\\\
2(3)+3(4) &=18\\\\
6+12&=18
\end{aligned}

The statement is correct, so we know that our line is drawn correctly.

Draw the graph of 4x-2y=20.

**Substitute ** \bf{x=0} ** to find the ** \textbf{y} **-intercept.**

Substitute x=0 into 4x-2y=20.

\begin{aligned}4x-2y &= 20 \\\\
4(0)-2y &=20\\\\
-2y&=20\\\\
y&=-10
\end{aligned}

This means the line crosses the y -axis at -10. The y -intercept is -10.

**Substitute ** \bf{y=0} ** to find the ** \textbf{x} **-intercept.**

Substitute y=0 into 4x-2y=20.

\begin{aligned} 4x-2y &= 20 \\\\
4x-2(0) &=20\\\\
4x&=20\\\\
x&=-5
\end{aligned}

This means the line crosses the x -axis at 5. The x -intercept is 5.

**Join the two points, then substitute the values of a third point to check your answer.**

Join the two points with a straight line:

Always check your answer by substituting the values of a third point into the original equation.

The point (4,-2) lies on the line we have drawn. We substitute x=4, \ y=-2 into the equation 4x-2y=20 to check.

\begin{aligned} 4x-2y &= 20 \\\\
4(4)-2(-2) &=20\\\\
16+4&=20
\end{aligned}

The statement is correct, so we know that our line is drawn correctly.

**Take care with negative values**

It is easy to make mistakes when substituting negative values into algebraic expressions.

For example, substituting x=-2 into y=x^{2}+3 will be:

y=(-2)^2+3=4+3=7 \ \color{green} \textbf{✔}

NOT

y=-2^2+3=-4+3=-1 \ \color{red} \textbf{✘}

**Graphs of quadratic functions have symmetry**

Graphs of quadratic functions have a line of symmetry. This can be useful as a check on whether you have plotted the points correctly.

**Turning points should be drawn carefully**

Quadratic graphs and cubic graphs have turning points. They should be drawn with care so they are not very flat, nor very pointy.

1. Complete the table of values for y=5x-4.

The missing y -coordinates can be found by substituting the x -values into y=5x-4.

When x=-1, \ y=5\times -1-4=-5-4=-9

When x=0, \ y=5\times 0-4=0-4=-4

When x=2, \ y=5\times 2-4=10-4=6

2. Draw the graph of y=4-2x.

The graph of y=4-2x is a straight-line going through these points.

The points should be joined with a straight-line.

3. Draw the graph of 4x+3y=24.

We can work out where the line crosses the axes.

To find where the line crosses the y -axis, substitute x=0.

\begin{aligned}
4x+3y &= 24 \\\\
4(0)+3y &=24\\\\
3y&=24\\\\
y&=8
\end{aligned}

This means the line crosses the y -axis at 8.

To find where the line crosses the x -axis, substitute y=0.

\begin{aligned}
4x+3y &= 24 \\\\
4x+3(0) &=24\\\\
4x&=24\\\\
x&=6
\end{aligned}

This means the line crosses the x -axis at 6.

4. Complete the table of values for y=x^{2}+2x.

The missing y -coordinates can be found by substituting the x -values into y=x^{2}+2x.

When x=-2, \ y=(-2)^2 +2(-2)=4-4=0

When x=1, \ y=(1)^2 +2(1)=1+2=3

When x=2, \ y=(2)^2 +2(2)=4+4=8

5. Draw the graph of y=x^{2}-3.

The graph of y=x^{2}-3 is a curve going through these points.

The points should be joined with a smooth curve.

6. Complete the table of values for y=4^{x}.

The missing y -coordinates can be found by substituting the x -values into y=4^{x}.

When x=-1, \ y=4^{-1}=0.25

When x=1, \ y=4^1=4

When x=2, \ y=4^2=16

1. (a) Complete the table of values for y=3x-5.

(b) On the grid draw the graph of y=3x-5, for the values of x from -1 to 3.

**(4 marks)**

Show answer

(a)

For two correct y -values.

**(1)**

For all four correct y -values.

**(1)**

(b)

For at least 4 points correctly plotted.

**(1)**

For graph fully drawn correctly.

**(1)**

2. (a) Complete the table of values for y=x^{2}+3x-1.

(b) On the grid draw the graph of y=3x-5, for the values of x from -3 to 3.

**(4 marks)**

Show answer

(a)

For two correct y -values.

**(1)**

For all four correct y -values.

**(1)**

(b)

For at least 6 points correctly plotted.

**(1)**

For graph fully drawn correctly.

**(1)**

3. (a) Complete the table of values for y=\frac{12}{x}.

(b) On the grid draw the graph of y=\frac{12}{x}, for the values of x from 1 to 12.

**(4 marks)**

Show answer

(a)

For two correct y -values.

**(1)**

For all four correct y -values.

**(1)**

(b)

For at least 4 points correctly plotted.

**(1)**

For graph fully drawn correctly.

**(1)**

You have now learned how to:

- Plot straight-line graphs
- Plot graphs of functions including quadratic functions, cubic functions and exponential functions

- Inequalities on a graph
- Transformations of graphs
- Vectors
- Trigonometry
- Graphs of trigonometric functions

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