Question

I still need help with this question!



The average annual income, I, in dollars of a lawyer with an age of x years is modeled by the following function:

I= -425x^2+45,500x-650,000.

According to this model, what is the maximum average annual income, in dollars, a lawyer can earn?

I've heard that you need to change it to the form a(x+b)^2+c, but I'm not sure where to plug the numbers in for that equation, and I'm not even sure if that's correct. Could someone please help me??

Answer 1

Just find the vertex. The y coordinate of the vertex is the maximum.

Answer 2

The x coordinate of the vertex is -b/2a
Find that and substitute it in place of x and find the corresponding y.

Answer 3

http://www.wolframalpha.com/input/?i=+find+the+vertex+of%3A++y%3D+-425x^2%2B45%2C500x-650%2C000.

Answer 4

I ended up trying the quadratic formula..
So I did...
x=-b+/-√ b^2-4ac/2a
And got
x=-45,500+/-√ -45,500-4(-425)(-650,000)/2 (-425)
Which, I somehow worked out to equal: 1,300,000.
Or if I plug it in to the solution at the bottom of the page you gave me a link to, I get -54 (rounded up) as a result of x, plug it in, and get -4,346,300.

But I feel like I'm still going wrong somewhere? @Mertsj

Answer 5

The form would be ax^2 + bx + c

here we have a = -425, b = 45500, c = -650000

You don't want to use the quadratic formula — that solves the equation for values of x that make the function 0. In other words, the points where the parabola describing the average income crosses the x-axis, which is not the vertex.

You can complete the square on the formula to put it in vertex form and read off the coordinates of the vertex directly, or you can rely on the fact that if put in standard form (see above), the x coordinate of the vertex is given by x = -b/(2a)

I'll do both ways:

x = -b/(2a) = -45500/(2*-425) = -45500/-850 = 910/17 or about 53.5294

now we plug that into the formula for the average income to find just what it is at that point:

I = -425(53.5294)^2 + 45500(53.5294) - 650000 =

Answer 6

When I plug that in, I get 3003381.282! Would that be my final answer? :) @whpalmer4

Answer 7

hmm, you're not doing it correctly, I think.

what is 53.5294^2 * -425?

what is 45500 * 53.5294?

add the first two results, subtract 650000, what do you get?

Answer 8

Never mind, I calculated it again and got 567794.1176! I was forgetting the negative sign in front of 425! So my final answer would be: 567,794.1176? @whpalmer4

Answer 9

yes, though I think you could probably round that to the nearest penny, or even dollar :-)

Answer 10

Thank you so much for your help!!!!

Answer 11

completing the square on this problem is actually kind of messy — if you want a demo of how to do it, I'm happy to do it with a different equation.

Answer 12

If you could that would be amazing!! @whpalmer4

Answer 13

okay, let's say that our function for which we want to find the vertex is

y = -x^2 - 4x + 15 (just pulling random numbers out of the air)

in standard form, as we have written it, that's a = -1, b = -4, c = 15
a>0 implies that the parabola opens upward, like a bowl.
a<0 implies that the parabola is an inverted bowl.

Obviously, for there to be a maximum, we need the inverted bowl.

we want to get to the form

y = a(x-h)^2 + k

now, if we have (x-h)^2, that is (x-h)(x-h) = x^2 -hx -hx + h^2 = x^2 - 2hx + h^2
that's the key to "completing the square" — we need to figure out what kind of a fudge factor we need to add so we can write -x^2 - 4x + 15 as -1(x-h)^2 + k
hopefully it is evident we'll have -1(x^2+4x + <something>) + <something else>
and that will then turn into -1(x-h)^2 + k

to do this, we look at the value of the coefficient of the x term (b). we divide it in half and square it. that's the value that is h^2 in our (x-h)^2.

we know that for
x^2 - 2hx + h^2
to match up with
x^2 +4x + <something>

-2hx = 4x

right?

Answer 14

it's just making each piece look like the corresponding piece from the other equation

Answer 15

so if
-2hx = 4x
divide both sides by x
-2h = 4
h = -2

so our (x-h)^2 part will be (x-(-2))^2 = (x+2)^2 = x^2 + 4x + 4

now, we can't just add that +4 to our equation — that would change one side without changing the other. we have two choices: we can either add 4 to both sides, or we can both add and subtract 4 from the same side.

y = <stuff>
y + 4 = <stuff> + 4

or

y = <stuff>
y = <stuff> + 4 - 4

Answer 16

we have

y = -1(x^2 + 4x - 15)

y = -1(x^2 + 4x + 4 - 4 - 15)
Now x^2 + 4x + 4 = (x+2)^2, so we can rewrite that as
y = -1((x+2)^2 - 4 - 15)
y = -1((x+2)^2 - 19)
y = -1(x+2)^2 + 19

and that's our vertex form, with a = -1, h = -2, k = 19

so our vertex is at (-2, 19)

if we do it the other way, as a check:

a = -1, b = -4, c = 15
x = -(-4)/(2*-1) = 4/-2 = -2
so x coordinate is -2
y coordinate is y = -(-2)^2 -4(-2) + 15 = -4 + 8 + 15 = 19
so vertex is at (-2, 19)
which agrees with our previous result.

Answer 17

doing that process on the original problem ends up with manipulating fractions like 910/17 and 828100/289 :-)

Answer 18

Thank you so much! I understand so much more now!! @whpalmer4

Answer 19

It's one of those things where practice helps considerably!

Here's a graph of the parabola from the original problem: