I need help! Math.

Question

I need a written response.

 

The chsnge in water vapor in a cloud is modeled by a polynomial function, C(x). Describe how to find the X-intercepts of C(x) and how to construct a rough graph kt C(x) so that the meteorilogist can predict when there will be no change in the water vapor. You may create a sample polynomial of degree 3 or higher to use in your explanation.

Answer 1

Hi Kichi.

To find the x-intercepts of any function, set C(x)=0 and solve for x.   

A sample polynomial of degree 3 might be  C(x)=(x-1)(x+2)(x-3) = x³-2x²-5x+6

To find the x-intercepts, C(x)=0 so (x-1)(x+2)(x-3)=0, or x=1, x=-2, x=3

 

Is this a calculus class?  If so, you would find the minimums (valleys) and maximums (peaks) by taking the derivative of C(x) and setting the derivative to equal 0.

 

For our example function, C(x) = x³-2x²-5x+6, so the derivative would be

 

C'(x)=3x²-4x-5     Set C'(x)=0 to find the min and max.   

3x²-4x-5=0  when x=2.1196 and x=-0.7863

 

 

Now you can plot your x-intercepts along the x-axis.  Plug x=2.1196 into C(x) to find the y-coordinate of this min.   Plug x=-0.7863 into C(x) to find the y-coordinate of this min.

 

Now you know the general shape of an x-cubed function (up from y=-infinity on the left, crosses x-axis at -2, makes a peak at x=-0.7863, crosses the y-axis at y=6 (set x's=0), crosses the x-axis at x=1, makes the bottom of the valley at x=2.1196, on its way up to y=+infinity, it crosses the x-axis at x=3.

 

And that is how you make a sketch.  If all it has to be is a rough graph, come from bottom left up and be sure you cross the x-axis at -2, 1, and 3  and have  a max between x=-2 and x=1.  Be sure to cross the y-axis at 6.  Be sure to have a min between 1 & 3 and finish with the graph heading up/right forever more.