Two lines through the point(1,-3) are tangent to the curve y=x2. Find the equations of these two lines and make a sketch to verify your results. Thank you
To find the equations of the two tangent lines to the curve y = x^2 through the point (1, -3), we need to follow these steps:
1. Determine the slope of the tangent line at the point of tangency.
2. Use the slope and the point (1, -3) to find the equation of each tangent line.
3. Sketch the graph of the curve y = x^2 and the two tangent lines to verify the results.
Step 1: Determine the slope of the tangent line at the point of tangency.
The slope of the tangent line to a curve at a given point can be found by taking the derivative of the function representing the curve. For y = x^2, the derivative is dy/dx = 2x. Evaluate this derivative at x = 1 to get the slope at the point of tangency:
dy/dx = 2(1) = 2
So, the slope of the tangent line at the point (1, -3) is 2.
Step 2: Use the slope and the point (1, -3) to find the equation of each tangent line.
We can use the point-slope form of a line to find the equation, which is y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Tangent line 1:
Using the point-slope form and substituting x1 = 1, y1 = -3, and m = 2, we have:
y - (-3) = 2(x - 1)
y + 3 = 2x - 2
y = 2x - 5
So, the equation of the first tangent line is y = 2x - 5.
Tangent line 2:
Using the point-slope form and substituting x1 = 1, y1 = -3, and m = 2, we have:
y - (-3) = 2(x - 1)
y + 3 = 2x - 2
y = 2x - 5
So, the equation of the second tangent line is y = 2x - 5.
Step 3: Sketch the graph of the curve y = x^2 and the two tangent lines to verify the results.
Now, we can plot the graph of y = x^2 and the two tangent lines on a coordinate plane. The point (1, -3) will be on both tangent lines.
Here is an ASCII representation of the graph:
* (1,-3)
|
| . (2,4)
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| . (0,-3)
|
|
_____________|_____________________________________
-2 0 2 4 6 8 x
This sketch shows the parabolic curve y = x^2 and the two tangent lines. The tangent lines intersect the curve at the point (1, -3) and have slopes of 2.