### Graphs of quadratics

Consider the relationship given by y = x\(^2\). This is not a linear relationship because x is raised to the power 2. First, we complete a table of values.

\(x\) | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|---|---|

\(y = x^2\) | 9 | 4 | 1 | 0 | 1 | 4 | 9 |

We then plot the corresponding points and join them with a smooth curve to obtain the graph below:

This graph is called a **parabola**. The point (0, 0) is called the **vertex** or **turning point** of the parabola. The vertex corresponds to the minimum value of y, since x\(^2\) is positive except when x = 0.

The graph is symmetrical about the y-axis. This line is called the **axis of symmetry**.

#### Example 1

Plot the graph of y = 2 x\(^2\) − 1 for −2 ≤ x ≤ 2. Find the axis of symmetry and the coordinates of the vertex.

#### Solution

The table of values is shown:

\(x\) | –2 | –1 | 0 | 1 | 2 |
---|---|---|---|---|---|

\(y = 2 x^2 - 1\) | 7 | 1 | –1 | 1 | 7 |

Notice that the y-values are the same for x = −1 and x = 1. The equation of the axis of symmetry is x = 0. When x = 0, y = −1, so the vertex has coordinates (0, −1).

#### Example 2

Plot the graph of y = x\(^2\) − 4 for −3 \(\le\) x \(\le\) 3. Give the coordinates of the vertex, the equation of the axis of symmetry and the coordinates of the x-intercepts and the y-intercept.

#### Solution

The table of values is shown:

\(x\) | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|---|---|

\(y = x^2 - 4\) | 5 | 0 | −3 | −4 | −3 | 0 | 5 |

The vertex has coordinates (0, −4).

The equation of axis of symmetry is x = 0.

The coordinates of the y-intercept are (0, −4).

The coordinates of the x-intercepts are (−2, 0) and (2, 0).