How Do You Find Tangent Lines to a Parabola Through a Given Point?

[Lilly]

hi,
i know this problem has to do with derivatives and I'm sure its easy, but for some reason i just can't figure it out...
Find equations of both lines through the point (2,-3) that are tangent to the parabola y= x^2 + x

help please...i know the derivative is 2x + 1 ...and that the 2 equations will be linear and pass through the point above, obviously...just not sure where to go from there.

 

[Himura Kenshin]

This is differentiation, you have dy/dx =2x+1.then substitute the x=2 in the dy/dx,then you will find that dy/dx =5(that is the gradient of the tangent)


y-(-3) =3(x-2)
y+3 = 3x-6
y= 3x-9 (equation of the tangent)

The othe equation you mention may be the normal for the curve.Use formula

m1 x m2 =-1

then you can solve the question.

 

[wais]

Hi there,

Don't worry, derivatives can be tricky at first but with practice, you'll get the hang of it! Let's break down this problem step by step.

First, you're correct that the derivative of y = x^2 + x is 2x + 1. This is because the derivative of x^2 is 2x, and the derivative of x is 1.

Next, we want to find the equations of lines that are tangent to the parabola at the point (2,-3). This means that the lines will have the same slope as the parabola at that point.

To find the slope of the parabola at (2,-3), we can plug in x = 2 into the derivative 2x + 1. This gives us a slope of 5. So, the lines we are looking for will have a slope of 5.

Now, we can use the point-slope form of a line to find the equations. The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point. Plugging in m = 5 and (x1, y1) = (2,-3), we get:

y - (-3) = 5(x - 2) and y + 3 = 5x - 10

These are the equations of the two lines that are tangent to the parabola at (2,-3). You can simplify them if you'd like to get the final equations in slope-intercept form (y = mx + b).

I hope this helps! Don't be discouraged if you struggle with derivatives at first, just keep practicing and it will become easier. Good luck!