# Quadratics and Harder Graphs

## Quadratics and Harder Graphs Revision

**Quadratic Graphs and Other 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 fairly flat, e.g.

\textcolor{red}{2}x^3 + \textcolor{limegreen}{3}x^2 + \textcolor{blue}{x}

**Interpreting Roots and Intercepts**

The **roots **of a quadratic are the values of x where the quadratic equals 0, so if

y= ax^2+bx+c

then the roots of ax^2+bx+c are the values of x when y=0. When the quadratic is graphed, the roots are the points on the graph where it crosses the x-axis. These points are called the **x-intercepts**.

So if x_1 and x_2 are roots of the quadratic, then the point (x_1, 0) and (x_2, 0) are on the graph.

In this example, the roots of the quadratic can be interpreted graphically as the points that the curve crosses the x-axis. Giving us

(-3,0) and (3,0)

So the roots are x=3 and x = -3

The **y-intercept** is where the curve crosses the x-axis. In this case it is at the point

(0,-3)

So the y-intercept is y=-3

The graphical interpretation of the x and y-intercepts can also be done for other types of graphs such as cubic graphs.

**Reciprocal Graphs**

**Reciprocal graphs** have the general form

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

e.g.,

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

**Exponential Graphs**

**Exponential graphs** have the general form

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

e.g.,

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

**Example 1: 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:** Substitute our values of x into the equation to get the corresponding y values.

For example, when x=\textcolor{red}{-2}, we get

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

**Step 3:** Continue this process for all other values of x

**Step 4:** 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 graph.

**Example 2: 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:** Substitute our values of x into the equation to get the corresponding y values.

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

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

**Step 3:** Continue this process for all other values of x

**Step 4:** 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 graph.

**Example 3: Plotting Reciprocals**

Using the equation y= \dfrac{1}{x} draw a table of coordinates from x = -2.5 to x=2.5. Use the values to plot the graph between these x values.

**[3 marks]**

**Step 1: **Draw a table of the coordinates for x from -2.5 to 2.5

**Step 2:** Substitute our values of x into the equation to get the corresponding y values.

For example, for x = -2, we get

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

**Step 3:** Continue this process for all other values of x

**Step 4:** From the table we get coordinates to plot. e.g. (-2, -0.5)

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

**Example 4: Plotting Exponentials**

Using the equation y= 2^x 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:** Substitute our values of x into the equation to get the corresponding y values.

For example, for x = -1, we get

y= 2^-1 =0.5

**Step 3:** Continue this process for all other values of x

**Step 4:** From the table we get coordinates to plot. e.g. (-1, 0.5)

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

## Quadratics and Harder Graphs 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:** 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.