# Quadratic Graphs

## Quadratic Graphs Revision

**Quadratic Graphs**

Quadratic graphs can be **sketched** on a set of axes, and from here the **roots** can be found.

**Careful**: In some cases, the equation may need to be rearranged to the form of a quadratic.

**Sketching Quadratic Graphs and Finding the Roots**

Quadratic graphs take the form

y=x^2+bx+c

They have a general U shape, with one **line of symmetry** halfway between the x intercepts. The **x** intercepts are circled in **red**. The **y** intercept is circled in **blue**.

**Sketching:**

To sketch a quadratic you can create an xy table, for a selection of x values, and plot the coordinates. Connecting these points will create the shape of your quadratic.

**Roots: **

To find the roots of the quadratic equation, you take a look at your sketch and see where the graph crosses the x axis.

**Note:** The roots of a quadratic equation are the same as the x intercepts.

Take a look at the examples below.

**Example 1**

Plot the following graph on a set of x and y axes,

y=x^2+2x-8

Hence, find the roots of the equation.

**[3 marks]**

**Creating The xy Table**

Substituting the values x=-5 to x=3, we get the following table:

Plotting these points as coordinates we get the following graph (as seen on the right).

**Finding the Roots**

From the sketch we can see the graph crosses the x axis at -4 and 2. These are the roots.

**Example 2**

Plot the following graph on a set of x and y axes,

y=x^2+8x+15

Hence, find the roots of the equation.

**[3 marks]**

**Creating The xy Table**

Substituting the values x=-8 to x=0, we get the following table:

Plotting these points as coordinates we get the following graph (as seen on the right)

**Finding the Roots**

From the sketch we can see the graph crosses the x axis at -5 and -3. These are the roots.

## Quadratic Graphs Example Questions

**Question 1:** On the axes below plot the the following quadratic graph, hence also find the roots.

y=x^2+7x+10

**[4 marks]**

y=x^2+7x+10

**Creating The xy Table**

Substituting the values x=-7 to x=0, we get the following table:

Plotting these points as coordinates we get the following graph

**Finding the Roots**

From the sketch we can see the graph crosses the x axis at -5 and -2. These are the roots.

**Question 2:** On the axes below plot the the following quadratic graph, hence also find the roots.

y=x^2-9x+14

**[4 marks]**

y=x^2-9x+14

**Creating The xy Table**

Substituting the values x=0 to x=9, we get the following table:

Plotting these points as coordinates we get the following graph

**Finding the Roots**

From the sketch we can see the graph crosses the x axis at 2 and 7. These are the roots.

**Question 3:** On the axes below plot the the following quadratic graph, hence also find the roots.

y-x^2=2x+1

**[4 marks]**

This question we first need to rearrange the equation to the form of a quadratic:

y=x^2+2x+1

**Creating The xy Table**

Substituting the values x=-8 to x=0, we get the following table:

Plotting these points as coordinates we get the following graph

**Finding the Roots**

From the sketch we can see the graph touches the x axis at -1. For this particular question the quadratic equation only has one root (-1)

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