CPM Homework Help : CALC Problem 3-11

Below is the graph of the function $f(x) = 2x^3$ with tangents drawn at $x = −2$ , $−1$ , $1$ , and $2$ . Use the slopes provided in the graph to find the slope function $f'(x)$ . Notice that $f'(0) = 0$ . It might be helpful to make a table of data relating $x$ to $m$ . Slope at a Point eTool (Desmos)

Table:

$x$	$-2$	$-1$	$0$	$1$	$2$
$m$	$24$	$6$	$0$	$6$	$24$

The data in the table was taken from the graph.
' $x$ ' represents $x$ -values.
' $m$ ' represents slope of the tangent line.

Hint 1:

The original graph, $f(x)$ , is cubic. Does the table of slopes also appear to have a cubic pattern? If not, what type of function would model its pattern?

Hint 2:

The slopes have a quadratic pattern! Clearly, the data does not fit the parent quadratic equation: $y = x^2$ . Find a transformation of $y = x^2$ that models the data.

Use the eTool below to view the tangent lines.
Click on the link to the right to view the full version of the eTool: Slope at a Point eTool