Find the equations for the two tangent lines.

In summary, the equations for the two tangent lines on the graph f(x) = - (x-3)^2 - 4 through the point (2,5) are y = 2x+1 and y = 2x+9. To find the second tangent line, consider the lines from points (x,f(x)) on the curve that pass through (2,5) and then solve for the lines whose slope matches f'(x). It is important to note that the point (2,5) is not on the curve y = f(x) and the tangent line at x=2 does not pass through (2,5).
  • #1
Jinxypo
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Homework Statement


Find the equations for the two tangent line on the graph f(x) = - (x-3)^2 - 4 through the point (2,5)

Homework Equations


The Attempt at a Solution


I already solved for f '(x) which is -2x +6. Then I plug in 2 for f '(x) in order to find the slope, which is 2. Using the equation y - y1 = m(x -x1) I found the equation for my first tangent line to be y = 2x+1. My question is how do i find the second tangent line, and did I do something wrong?
 
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  • #2
The point (2,5) is not on the curve y = f(x). Furthermore, the tangent line to f(x) at x=2 does not pass through the point (2,5). y=f(x) is a parabola, you need to consider the lines from points (x,f(x)) on the curve that pass through (2,5) and then solve for the lines whose slope matches f'(x).
 

FAQ: Find the equations for the two tangent lines.

What does it mean to find the equations for two tangent lines?

Finding the equations for two tangent lines means determining the mathematical equations that represent the two lines that touch a curve at a specific point, known as the point of tangency. These equations can be used to calculate the slope and y-intercept of the tangent lines, which provide information about the behavior of the curve at that point.

How do I find the equations for two tangent lines?

To find the equations for two tangent lines, you must first identify the point of tangency on the curve. Then, you can use the slope formula to calculate the slope of the tangent lines. Next, use the point-slope formula to determine the equation of each tangent line. Finally, simplify the equations to their standard form, y = mx + b, where m is the slope and b is the y-intercept.

Can there be more than two tangent lines to a curve?

Yes, it is possible for a curve to have more than two tangent lines. This can occur at points where the curve has a sharp turn or a point of inflection. In these cases, there can be multiple tangent lines that touch the curve at the same point.

What is the significance of finding the equations for two tangent lines?

Finding the equations for two tangent lines can provide valuable information about the behavior of a curve at a specific point. The slope of the tangent lines can indicate the rate of change of the curve at that point, while the y-intercept can represent the initial value or starting point of the curve at that point.

Are there any shortcuts or tricks for finding the equations of tangent lines?

There are no shortcuts for finding the equations of tangent lines, but there are some tips that can make the process easier. One helpful tip is to use the derivative of the curve at the point of tangency to determine the slope of the tangent lines. Additionally, you can use the symmetry of the curve to quickly determine the equations of the tangent lines on either side of the point of tangency.

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