CPM Homework Help : APCALC Problem 3-11

Below is a graph of the function $f(x) = 2x^3$ with tangent lines drawn at $x = −2, −1, 1,$ and $2$ . Use the slopes provided in the graph to determine the slope function $f^\prime$ . Notice that $f^\prime (0) = 0$ . It might be helpful to make a table of data relating $x$ to $m$ .

Hint a:

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

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

Hint:

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:

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.

Increasing cubic curve, centered at the origin, with 4 tangents, labeled as follows: at the point (negative 2, comma negative 16), m =. 24, at the point (negative 1, comma negative 2), m = 6, at the point (1, comma 2), m = 6, & at the point (2, comma 16), m = 24.

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