Chapter 4.7, Problem 21PPS

To determine

To graph: the quadratic equation as mentioned in the problem and to write the quadratic equation in vertex form.

Answer to Problem 21PPS

The vertex form is $\left(-{\left(x-\frac{17}{2}\right)}^2+\frac{1089}{4}\right)$ and vertex is $\left(\frac{17}{2},\frac{1089}{4}\right)$ .

Explanation of Solution

Given information:

The quadratic equation is $y=-15x^2+255x+3000$ .

Formula:

For vertex form of quadratic equation we will use $y={\left(x-h\right)}^2+k$ where $\left(h,k\right)$ is the vertex.

Calculation:

Consider the quadratic equation $y=-15x^2+255x+3000\Rightarrow -x^2+17x-200$ .

Recall that for vertex form of quadratic equation we will use $y={\left(x-h\right)}^2+k$ where $\left(h,k\right)$ is the vertex.

Therefore $y=-x^2+17x+200\Rightarrow -\left(x^2-2*x*\frac{17}{2}+\frac{289}{4}-\frac{289}{4}-200\right)\Rightarrow -\left({\left(x-\frac{17}{2}\right)}^2-\frac{1089}{4}\right)$ .

Therefore the vertex form of the quadratic equation $y=-15x^2+255x+3000$ is $\left(-{\left(x-\frac{17}{2}\right)}^2+\frac{1089}{4}\right)$ and the vertex is $\left(\frac{17}{2},\frac{1089}{4}\right)$ .

Graph:

The graph mentioned below is in real number line of x axis and y axis whose vertex is $\left(\frac{17}{2},\frac{1089}{4}\right)\Rightarrow \left(8.5,272.25\right)$ .

  

Interpretation:

Consider the quadratic equation $y=-x^2+17x+200$ .

The mentioned quadratic equation above is in vertex form $y=a{\left(x-h\right)}^2+k$ where $\left(h,k\right)$ is the vertex.

Therefore the vertex of equation is $\left(\frac{17}{2},\frac{1089}{4}\right)\Rightarrow \left(8.5,272.25\right)$ .

We can write the mentioned quadratic equation in the standard form as mentioned below

  $y=-x^2+17x+200$ .

Which is comparable to $a{x}^2+bx+c$ where $a=-15$ which is less than $0$ .

Since $a<0$ the direction of graph will be negative side of y axis.

The discriminant of standard form of quadratic equation $y=-x^2+17x+200$ is $D=b^2-4*a*c\Rightarrow d={\left(17\right)}^2-4*\left(-1\right)*\left(200\right)=289+800=1089$ .

Here $D>0$ .

Therefore the graph will lie below x axis.

At $x=0$ the $y=-x^2+17x+200$ will yield $y=200$ .

Therefore graph will cut y axis at $\left(0,200\right)$ .

Therefore the graph will be as mentioned below.

.

Therefore the vertex form of the quadratic equation $y=-15x^2+255x+3000$ is $\left(-{\left(x-\frac{17}{2}\right)}^2+\frac{1089}{4}\right)$ and the vertex is $\left(\frac{17}{2},\frac{1089}{4}\right)\Rightarrow \left(8.5,272.25\right)$ .