## What you need to know

**Harder Graphs**

This topic includes graphs which are not straight lines.

These include, **quadratic graphs**, **cubic graphs**, **reciprocal graphs** and **exponential graphs**.

You will need to be able to identify and plot these graphs.

You need to be happy with the following topics:

**Quadratic Graphs**

Quadratic graphs have the general form

\textcolor{red}{a}x^2 + \textcolor{limegreen}{b}x+\textcolor{blue}{c}

These form a \bigcup or \bigcap shape, examples are shown below:

Note: \textcolor{limegreen}{b} and \textcolor{blue}{c} can be zero, as is the case with y=x^2

**Cubic Graphs **

Cubic graphs have the general form

\textcolor{red}{a}x^3 + \textcolor{limegreen}{b}x^2+\textcolor{blue}{c}x +\textcolor{maroon}{d}

These form S shape in the middle.

Note: Sometimes this S can be flat.

**Reciprocal Graphs**

Reciprocal functions have the general form

y = \dfrac{\textcolor{red}{k}}{x}

e.g:

y = \dfrac{\textcolor{red}{1}}{x}

**Exponential Graphs**

Exponential functions have the general form

y = \textcolor{blue}{k}^x

e.g:

y = \textcolor{blue}{3}^x

**Example: Plotting Quadratics **

Plot the following quadratic equation:

y=x^2-x-5

**[2 marks]**

First draw a table of coordinates from x=-2 to x=3, then use the values to plot the graph between these values of x.

**Step 1:** Draw a table for the values of x between -2 and 3.

**Step 2:** Next we need to substitute our value of x into the equation to get the corresponding y value.

E.g: When x=\textcolor{red}{-2}, we get

y=(\textcolor{red}{-2})^2-(\textcolor{red}{-2})-5=4+2-5= \textcolor{blue}{1}.

**Step 3:** We then continue this process for all other values of x

**Step 4:** Then from the table we get coordinates to plot. e.g. (\textcolor{red}{-2}, \textcolor{blue}{1})

Once plotted, we join all the points with a **smooth curve **giving the following.

**Example: Plotting **Cubics

Using the equation y=x^3-2x^2, draw a table of coordinates from x=-1 to x=3. Use the values to plot the graph between these x values.

**[3 marks]**

**Step 1: **Draw a table of the coordinates for x from -1 to 3

**Step 2:** Next we need to substitute our value of x into the equation to get the corresponding y value.

For example, for x=\textcolor{red}{1}, we get

y=\textcolor{red}{1}^3-2(\textcolor{red}{1})^2=\textcolor{blue}{-1}.

**Step 3:** We then continue this process for all other values of x

**Step 4:** Then from the table we get coordinates to plot. e.g. (\textcolor{red}{1}, \textcolor{blue}{-1})

Once plotted, we join all the points with a **smooth curve **giving the following.

### Example Questions

**Question 1:** Using the equation y=x^2+4x-9, complete the table of coordinates below. Use these coordinates to plot the graph between x=-5 and x=2.

**[2 marks]**

We will complete this table by substituting in the values of x to get the missing values of y. For example, when x=2,

y=(-4)^2+4(-4)-9=16-16-9=-9

Continuing this with the rest of the x values, we get the completed table below.

Then, plotting these coordinates on a pair of axes and joining them with a curve, we get the graph below.

**Question 2:** Using the equation y=x^3+3x^2-4, complete the table of coordinates below. Use these coordinates to plot the graph between x=-4 and x=1.

**[2 marks]**

We will complete this table by substituting in the values of x to get the missing values of y. For example, when x=-2,

y=(-2)^3+3(-2)^2-4=-8+12-4=0

Continuing this with the rest of the x values, we get the completed table below.

Then, plotting these points on a pair of axes and joining them with a curve, we get the graph below.

**Question 3:** (HIGHER ONLY) Using the equation y=(0.2)^x, complete the table of coordinates below. Use these coordinates to plot the graph between x=-1 and x=4.

**[2 marks]**

We will complete this table by substituting in the values of x to get the missing values of y. For example, when x=2,

y=(0.2)^2=0.04

Continuing this with the rest of the x values, we get the completed table below.

Then, plotting these points on a pair of axes (to the best of your ability – some of the y values are so small they’re going to end up practically on the x-axis) and joining them with a curve, we get the graph below.

**Question 4:** Using the equation y=2^x, draw a table of coordinates from x=-3 to x=2.

Use the values to plot the graph between these x values.

**[2 marks]**

We draw this table by substituting the x values into the equation. For example, for x=1, we get

y=2^1=2.

Carrying this on with the rest of the numbers, we get the table above. Then, plotting these points and joining them with a curve, we get the graph to the right.

The exponential graph also has an asymptote along the x-axis. Its shape varies very little, except that when the base of the exponential (here, the function is 2^x so the base is 2) is a number between 0 and 1, the shape of the graph is a mirror image of this one. Specifically, a reflection in the y-axis.

**Question 5:** Using the equation y=\dfrac{1}{x}, draw a table of coordinates from x=1 to x=5. Use the values to plot the graph between x=1 and x=5.

**[2 marks]**

We draw this table by substituting the x values into the equation. For example, for x=2, we get

y=\dfrac{1}{2}=0.5.

Then, plotting these points on a pair of axes and joining them with a curve, we get the graph below.

### Worksheets and Exam Questions

#### (NEW) Plotting Quadratics and Harder Graphs Exam Style Questions

Level 5-6#### Drawing Quadratic Graphs

Level 4-5#### Plotting Quadratics and Harder Graphs

Level 4-5#### Plotting Harder graphs

Level 6-7### Learning resources you may be interested in

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