Question

find the range of the quadratic function write your answer as an inequality using x or y as appropriate. or you may instead click on empty set or all realsas the answer

Answer 1

Step-by-step explanation

The range of a function is a set composed of all of its output values. In this case this means all the possible y-values that the function can take. This is a cuadratic function which means that it has either a minimum or a maximum y-value that delimits its range. The sign of its leading coefficient is - which implies that this function has a maximum. This maximum is also known as the vertex of the function and if it's the point (h,k) then the range of this function can be described with this inequality:

`y\leq k`

So if we find the y-value of the vertex we find the range. In order to find it we could find its x-value first. If the function has two x-intercepts then the x-value of the vertex is the midvalue of these two. Then we should find the x-intercepts so we find the x-value of the vertex and with it its y-value.

For any given quadratic equation of the form ax²+bx+c=0 its x-intercepts are given by:

`r=(-b\pm√(b^2-4ac))/(2a)`

In our case we have a=-3, b=30 and c=-72 so we get:

`\begin{gathered} r=(-30\pm√(30^2-4\cdot(-3)\cdot(-72)))/(2\cdot(-3))=(-30\pm√(36))/(-6)=(-30\pm6)/(-6) \\ r=(-30+6)/(-6)=4\text{ and }r=(-30-6)/(-6)=6 \end{gathered}`

So the x-values of the x-intercepts are 4 and 6. Then the x-value of the vertex (h) of this function is:

`h=(4+6)/(2)=(10)/(2)=5`

Then the y-value of the vertex k is given by taking x=5 in the equation:

`k=-3h^2+30h-72=-3\cdot5^2+30\cdot5-72=3`Answer

Then the answer is:

`y\leq3`