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

In summary: The slope of the line is 2x and the x coordinate of the point where the line intersects the parabola is (0, -4).
  • #1
HHenderson90
9
0

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.
 
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  • #2
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?
 
  • #3
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.
 
  • #4
HHenderson90 said:
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.
 
  • #5
This helped a lot! I got the answer.
 

1. What is a tangent line?

A tangent line is a line that touches a curve at one point, without intersecting it. It represents the instantaneous slope of the curve at that point.

2. How do you find the point where a tangent line hits a curve?

To find the point where a tangent line hits a curve, you need to have a point on the curve and the slope of the curve at that point. Then, you can use the point-slope formula to find the equation of the tangent line and solve for the x and y values of the point of intersection.

3. Can you find the point of tangency if the given point is not on the curve?

No, the point of tangency must be on the curve in order to find it. If the given point is not on the curve, you will need additional information about the curve, such as its equation, in order to find the point of tangency.

4. What happens when there are multiple points of tangency on a curve?

If there are multiple points of tangency on a curve, it means that the curve has multiple points where the slope is equal to the slope of the tangent line. In this case, you will need to use the derivative of the curve to find the points of tangency.

5. Is there a shortcut or formula to find the point of tangency?

Yes, there is a formula known as the derivative formula, which involves finding the derivative of the curve, plugging in the given point, and solving for the x and y values of the point of tangency. However, it is important to have a good understanding of the concept of tangent lines before using this formula.

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