# 5.1 Parabolic functions

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## Investigation : average gradient - parabolic function

Fill in the table by calculating the average gradient over the indicated intervals for the function $f\left(x\right)=2x-2$ :

 ${x}_{1}$ ${x}_{2}$ ${y}_{1}$ ${y}_{2}$ $\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ A-B B-C C-D D-E E-F F-G

What do you notice about the average gradient over each interval? What can you say about the average gradients between A and D compared to the averagegradients between D and G?

The average gradient of a parabolic function depends on the interval and is the gradient of a straight line that passes through the points on the interval.

For example, in [link] the various points have been joined by straight-lines. The average gradients between the joined points are then the gradients of the straight lines that pass through the points.

Given the equation of a curve and two points ( ${x}_{1}$ , ${x}_{2}$ ):

1. Write the equation of the curve in the form $y=...$ .
2. Calculate ${y}_{1}$ by substituting ${x}_{1}$ into the equation for the curve.
3. Calculate ${y}_{2}$ by substituting ${x}_{2}$ into the equation for the curve.
4. Calculate the average gradient using:
$\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

Find the average gradient of the curve $y=5{x}^{2}-4$ between the points $x=-3$ and $x=3$

1. Label the points as follows:

${x}_{1}=-3$
${x}_{2}=3$

to make it easier to calculate the gradient.

2. We use the equation for the curve to calculate the $y$ -value at ${x}_{1}$ and ${x}_{2}$ .

$\begin{array}{ccc}\hfill {y}_{1}& =& 5{x}_{1}^{2}-4\hfill \\ & =& 5{\left(-3\right)}^{2}-4\hfill \\ & =& 5\left(9\right)-4\hfill \\ & =& 41\hfill \end{array}$
$\begin{array}{ccc}\hfill {y}_{2}& =& 5{x}_{2}^{2}-4\hfill \\ & =& 5{\left(3\right)}^{2}-4\hfill \\ & =& 5\left(9\right)-4\hfill \\ & =& 41\hfill \end{array}$
3. $\begin{array}{ccc}\hfill \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}& =& \frac{41-41}{3-\left(-3\right)}\hfill \\ & =& \frac{0}{3+3}\hfill \\ & =& \frac{0}{6}\hfill \\ & =& 0\hfill \end{array}$
4. The average gradient between $x=-3$ and $x=3$ on the curve $y=5{x}^{2}-4$ is 0.

## Summary

• Average gradient of straight line

## End of chapter exercises

1. An object moves according to the function $d=2{t}^{2}+1$ , where $d$ is the distance in metres and $t$ the time in seconds. Calculate the average speed of the object between 2 and 3 seconds. The speed is the gradient of the function $d$
2. Given: $f\left(x\right)={x}^{3}-6x$ . Determine the average gradient between the points where $x=1$ and $x=4$ .

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