Creating a Function: Create a graph of a function given the following information...

[RGeisler]

This is more like pre-calculus but I'm just looking for some help

1. Create a graph of a function given the following information:
   • The instantaneous rate of change at x= 2 is zero.
   • The instantaneous rate of change at x = 3is negative.
   • The average rate of change on the interval 0<x<4 is zero. (let the < sign represent greater than or equal to and less than or equal to)

I understand what the graph should look like, but not what steps to take to achieve a function that is momentarily parallel

 

[mmm4444bot]

I think you only need to draw a rough sketch. (Many different graphs are possible.)

They do not ask for an algebraic formula. Are you trying to use an app, to plot the graph?

You have an idea about the shape of the curve, so draw it! Draw your curve from x=0 to x=4. Now confirm that your graph meets all three conditions.

At x = 2 on your function's curve, does the tangent line have slope zero?

Does the tangent line at x = 3 have negative slope?

Average rate of change on interval [a,b] is zero:

[ f(b) - f(a) ] / [b - a] = 0

Do the (x,y) coordinates from the endpoints of your graph satisfy this equation?

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I'm not sure that I understand "momentarily parallel". Are they talking about a tangent line that moves along the curve, as x increases? The tangent line is parallel to the x-axis momentarily, as it passes around the turning point on the curve. The tangent line's slope is zero, at turning points. Here's an example. Slide the blue dot, to move the tangent line back and forth (or click the play button).