### Home > CALC > Chapter 3 > Lesson 3.1.1 > Problem3-11

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)

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

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

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?

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