Chapter 4.7, Problem 33PPS

To determine

To graph: the quadratic equation as mentioned in the problem.

Explanation of Solution

Given information:

The quadratic equation is $y=-4{\left(x-10\right)}^2-10$ .

Graph:

The graph mentioned below is in real number line of x axis and y axis.

Where vertex $V\left(10,-10\right)$ .

  

Interpretation:

Consider the quadratic equation $y=-4{\left(x-10\right)}^2-10$ .

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 $\left(10,-10\right)$ .

We can write the mentioned quadratic equation in the standard form as $y=-4x^2+80x-410$ which is comparable to $a{x}^2+bx+c$ where $a=-4$ 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 $y=-4x^2+80x-410$ is $D=b^2-4*a*c\Rightarrow d={\left(80\right)}^2-4*\left(-4\right)*\left(-410\right)=6400-6560=-160$ .

Here also $D=-160<0$ .

Therefore the graph will lie below the x axis.

At $x=0$ the $y=-4x^2+80x-410$ will yield $y=-410$ .

Therefore graph will cut y axis at $\left(0,-410\right)$ .

Therefore the graph will be as mentioned below.