5.1 Parabolic functions

Parabolic functions

Investigation : average gradient - parabolic function

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

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.

Method: average gradient

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:
   $$\frac{y_2-y_1}{x_2-x_1}$$

Summary

• Definition of average gradient
• Average gradient of straight line
• Average gradient of parabola

End of chapter exercises

1. An object moves according to the function $d=2t^2+1$ , 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 $x=4$ .