Mathematics » Functions I » Quadratic Functions

# Functions of the Form y = ax2 + q

## Optional Investigation: The effects of $$a$$and $$q$$ on a parabola.

Complete the table and plot the following graphs on the same system of axes:

1. $${y}_{1}={x}^{2}-2$$

2. $${y}_{2}={x}^{2}-1$$

3. $${y}_{3}={x}^{2}$$

4. $${y}_{4}={x}^{2}+1$$

5. $${y}_{5}={x}^{2}+2$$

 $$x$$ $$-\text{2}$$ $$-\text{1}$$ $$\text{0}$$ $$\text{1}$$ $$\text{2}$$ $${y}_{1}$$ $${y}_{2}$$ $${y}_{3}$$ $${y}_{4}$$ $${y}_{5}$$

Use your results to deduce the effect of $$q$$.

Complete the table and plot the following graphs on the same system of axes:

1. $${y}_{6}=-2{x}^{2}$$

2. $${y}_{7}=-{x}^{2}$$

3. $${y}_{8}={x}^{2}$$

4. $${y}_{9}=2{x}^{2}$$

 $$x$$ $$-\text{2}$$ $$-\text{1}$$ $$\text{0}$$ $$\text{1}$$ $$\text{2}$$ $${y}_{6}$$ $${y}_{7}$$ $${y}_{8}$$ $${y}_{9}$$

Use your results to deduce the effect of $$a$$.

## The effect of $$q$$

The effect of $$q$$ is called a vertical shift because all points are moved the same distance in the same direction (it slides the entire graph up or down).

• For $$q>0$$, the graph of $$f(x)$$ is shifted vertically upwards by $$q$$ units. The turning point of $$f(x)$$ is above the $$y$$-axis.

• For $$q<0$$, the graph of $$f(x)$$ is shifted vertically downwards by $$q$$ units. The turning point of $$f(x)$$ is below the $$y$$-axis.

## The effect of $$a$$

The sign of $$a$$ determines the shape of the graph.

• For $$a>0$$, the graph of $$f(x)$$ is a “smile” and has a minimum turning point at $$(0;q)$$. The graph of $$f(x)$$ is stretched vertically upwards; as $$a$$ gets larger, the graph gets narrower.

For $$0<a<1$$, as $$a$$ gets closer to $$\text{0}$$, the graph of $$f(x)$$ gets wider.

• For $$a<0$$, the graph of $$f(x)$$ is a “frown” and has a maximum turning point at $$(0;q)$$. The graph of $$f(x)$$ is stretched vertically downwards; as $$a$$ gets smaller, the graph gets narrower.

For $$-1<a<0$$, as $$a$$ gets closer to $$\text{0}$$, the graph of $$f(x)$$ gets wider.

 $$a<0$$ $$a>0$$ $$q>0$$ $$q=0$$ $$q<0$$
Table: The effect of $$a$$ and $$q$$ on a parabola.

You can use this Phet simulation to help you see the effects of changing $$a$$ and $$q$$ for a parabola.

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