Question

at what value of x is the tangent line to
y=x^2 + 3x +5 perpendicular to the line x-2y = 5

Answer 1

@SolomonZelman

Answer 2

you are pretty much given the slope of your tangent

Answer 3

\(\large\color{black}{x-2y = 5}\)
\(\large\color{black}{-2y+x = 5}\)
\(\large\color{black}{-2y= -x+5}\)
\(\large\color{black}{y= (1/2)x+5/2}\)

if a slope of a line that is perpendicular to yours is 1/2, what is the slope of your line?

Answer 4

1/2

Answer 5

no, 1/2 is a slope of a perpendilar line to yours.

A perpendicular slope (mathematicians excuse me please) to slope m is, -1/m.

Answer 6

I am not understanding this... if you have y= 1/2x +5/2 isn't the slope 1/2

Answer 7

yes. but this is the rearranged version of a line that is perpendicular to your tangent that you are finding.

Answer 8

I want you to give me a slope of a line that is perpendicular to the line with slope of 1/2.

Answer 9

2?

Answer 10

what is the sign?

Answer 11

ugh... I am not sure

Answer 12

positive ?

Answer 13

-2, no?
m -> -1/m
1/2 -> -2/1 -> -2

Answer 14

oh I see what you mean with -1/m now , ok

Answer 15

Yes, and now tell me what is the derivative of
\(\large\color{black}{y=x^2 + 3x +5}\)

Answer 16

2x +3

Answer 17

yes

Answer 18

so I plug in -2 as x?

Answer 19

the derivative of the function is the slope of it.
So therefore, knowing that the slope of the tangent is -2.
then you set f'(x)=-2.

Answer 20

\(\large\color{black}{f'(x)=2x+3}\)
\(\large\color{black}{-2=2x+3}\)

Answer 21

and that is the equation or would you continue ?

Answer 22

i mean no , I am not looking for an equation I am looking for the value of x so I continue to find x correct?

Answer 23

no that is not the equation, you are trying to find the x-coordinate

Answer 24

x =?

Answer 25

-2/5

Answer 26

no wait

Answer 27

-5/2

Answer 28

right?

Answer 29

yes

Answer 30

now find f(5/2)

Answer 31

omg yay ... I get it

Answer 32

ok , so now I plug that in to the y= equation right?

Answer 33

yes you need the f(5/2) found it?

Answer 34

I mean f(-5/2)

Answer 35

15/4

Answer 36

yes, that is what I am getting to

Answer 37

use poitn slope formula

Answer 38

verification.
https://www.desmos.com/calculator/3nt7l5veti

Answer 39

so the vlue of x is 15/4?

Answer 40

no that is the f(-5/2)

Answer 41

that is the y value of the point (on a function), where the x=-5/2.

Answer 42

http://www.wolframalpha.com/input/?i=%28-5%2F2%29%5E2%2B3%28-5%2F2%29%2B5

Answer 43

so what do I do now?

Answer 44

you plug in your point, (-5/2 , 15/4) and slope -2, into the point slope formula.

Answer 45

as I did in my desmos attachmentr

Answer 46

yeah that is what I got

Answer 47

yes, and you can see that it is correct. since it is tangent according to the graph.
https://www.desmos.com/calculator/zxm7radxgc

Answer 48

I just changed the color in this link

Answer 49

so I am done with the problem right?

Answer 50

questions about this problem?

Answer 51

I got to go now, it is 10:30 pm

Answer 52

see you!

Answer 53

thank you for your help !!!