Question

Answer the questions below about the quadratic function.g(×)=-×^2+10×-27Does the function have a minimum or maximum value? minimum or maximum Where does the minimum or maximum value occur?x=?what is the functions minimum or maximum value?

Answer 1

To find if this quadratic function has a minimum or a maximum value, we need to see the quadratic coefficient of the function. If the value of this coefficient is negative, the function will have a maximum. If the coefficient is positive, the quadratic function will have a minimum.

Therefore, in this case, we have:

$g(x)=-x^2+10x-27$

We can see that "in front of x^2, we have that the coefficient is negative. Therefore, this function has a maximum.

To find where this maximum occurs, we can use the formula for the vertex of a parabola (quadratic function):

$x_v=-\frac{b}{2a},y_v=c-\frac{b^2}{4a}$

We can get a, b, and c from the quadratic function in question:

$ax^2+bx+c\Rightarrow-x^2+10x-27$

Then, we have that:

a = -1

b = 10

c = -27

Then, to find the value of x when the maximum occurs, we have:

$x_v=-\frac{10}{2(-1)}=-\frac{10}{-2}=5\Rightarrow x_v=5$

And now, to find the value of y when the maximum occurs, we also have:

$y_v=-27-\frac{(10)^2}{4\cdot(-1)}=-27-\frac{100}{-4}=-27+25=-2\Rightarrow y_v=-2$

Therefore, the value for x is equal to 5 (x = 5), and the value for y is equal to -2 ( y = -2) when the quadratic function takes its maximum value.

A graph for this function can be seen as follows:

In summary, the function has a maximum value since the quadratic coefficient is negative, and the x-coordinate for this maximum is x = 5, and the y-coordinate for this maximum is y = -2 (and this is the maximum value for the function).