what is the maximum y value on the graph of y=f(x) f(x)= -x^2+6x+8

Question

what is the maximum y value on the graph of y=f(x) f(x)= -x^2+6x+8

Answer 1

You can graph the equation and see the maximum is infinity.

 

The equation, x^2 +6x + 8 represents a parabola, with vertex at (-3, -1), which is the minimum, sloping upwards for all points where x not equal to -3; it slopes upwards reaching infinity (or indeterminate).

 

 

Answer 2

Hi Kristi,

The easiest way (using algebra) to find the maximum value is to find the coordinates of the vertex. Because the coefficient on x² is negative, you know that the parabola opens downward. This means the vertex is a maximum (as opposed to a minimum if it opened upward).

How do you find the coordinates of the vertex? You complete the square.

 

First, factor out the coefficient on x². In this case, it's -1. Leave the constant term alone, and outside the parentheses.

 

f(x) = -(x²-6x) +8

Now, take the coefficient of x, divide by 2, then square it.

-6/2 = -3

(-3)^2 = 9

This is the number that you need to add to x^2-6x to make it a perfect square.

f(x) = -(x^2-6x+9) +8+9

Notice that the +9 inside the parentheses, multiplied by -1 in front, means I'm effectively subtracting 9 from the right hand side. That means I have to add 9 outside to balance it out.

Now, I rewrite the term in parentheses. 

f(x) = -(x-3)²+17

Notice that the second term in the parentheses will always be half the coefficient of x. (ex: x^2-8x... ---> (x-4)²

The general formula for a parabola is

y = -(x-h)²+k

where (h, k) are the x- and y-coordinates of the vertex, respectively.

It looks like k = 17, which means that the maximum value is 17.

Hope this helps!