How to find the point a tangent line hits when given a point off of the graph.

[HHenderson90]

Homework Statement

(a) Draw a diagram to show that there are two tangent lines to the parabola y = x^2 that pass through the point (0, -4). (Do this on paper. Your teacher may ask you to turn in this work.)

(b) Find the coordinates of the points where these tangent lines intersect the parabola.
( , ) (point with smaller x value)
( , ) (point with larger x value)

The Attempt at a Solution

I drew the graph of y=x^2, I also drew the point (0,-4) and I drew estimated tangent lines. I just don't understand how I go about finding the point where both of these tangent lines hit, I can estimate it but I know it's not looking for that.

I first got the derivative at 0 but realized that that is not the way to go about answering this.

 

[Dick]

Take (x,x^2) to be a point on your parabola. What's the slope of the tangent line there using the derivative? Now the line through (x,x^2) and (0,-4) has to have that same slope. How would you express that condition?

 

[HHenderson90]

Okay, so the slope would be 2x at (x,x^2) correct? I don't really understand that point though and where it exists on the graph, also how it has the same slope as that of (0,-4).
So, now that I know the slope I can figure out where the line hits the graph right? I just don't really know how to do that either honestly.

 

[Dick]

Okay, so the slope would be 2x at (x,x^2) correct? I don't really understand that point though and where it exists on the graph, also how it has the same slope as that of (0,-4).
So, now that I know the slope I can figure out where the line hits the graph right? I just don't really know how to do that either honestly.

What's the slope of the line through (x,x^2) and (0,-4)? It's change in y over change in x, right? That should equal 2x. Write that as an equation you can use to solve for x.

 

[HHenderson90]

This helped a lot! I got the answer.