Question

Consider the equation y\:=\:-x^2\:-\:7x\:+\:12. Determine whether the function has a maximum or a minimum value. State the maximum or minimum value. What are the domain and range of the function?

Answer 1

Answer:

Step-by-step explanation:

To answer this we need only know that negative parabolas are upside down, so by definition, it has a max point at the vertex. To find the vertex (h, k), the easy way to do this is to fill in the following expressions for h and k and solve:

$h=\frac{-b}{2a}$  and  $k=c-\frac{b^2}{4a}$ (These are derived from the quadratic formula). Filling in knowing our a = -1, b = -7, c = 12:

$h=\frac{-(-7)}{2(-1)}=\frac{7}{-2}=-\frac{7}{2}$  and

$k=12-\frac{(-7)^2}{4(-1)}=12-(\frac{49}{-4})=12+\frac{49}{4}=\frac{97}{4}$ so the vertex (aka max height occurs at $(-\frac{7}{2},\frac{97}{4})$. Depending upon what is meant by stating the max value, we may only need to state the k value (which is the same as the y coordinate, which is an up or down thing as opposed to the x value which is a side to side thing). The domain is all real numbers, as is the case for all x-squared parabolas, and the range is

R = {y | y ≤ 97/4}

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