Chapter 4.7, Problem 3E

a

To determine

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

Explanation of Solution

Given information:

The quadratic equation is $y=-3x^2-18x-26$ .

Graph:

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

Where vertex $\left(-3,1\right)$ .

  

Interpretation:

Consider the quadratic equation $y=-3x^2-18x-26$ .

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

We can write the mentioned quadratic equation in the vertex form as $\begin{array}{l}y=-3x^2-18x-26\Rightarrow y=-3\left(x^2+6x+\frac{26}{3}\right)\Rightarrow y=-3\left(x^2+2*x*3+9-9+\frac{26}{3}\right)\\ \Rightarrow y=-3\left({\left(x+3\right)}^2+\frac{26}{3}-9\right)\Rightarrow y=-3{\left(x+3\right)}^2+1.\end{array}$

Therefore the vertex form of the equation is $y=-3{\left(x+3\right)}^2+1$ .

Which is comparable to $y=a{\left(x-h\right)}^2+k$ where $a=-3$ which is less than $0$ .

Since $a<0$ the direction of graph will be negative side of y axis and the vertex is $\left(-3,1\right)$ .

The discriminant of standard form of quadratic equation $y=-3x^2-18x-26$ is $D=b^2-4*a*c\Rightarrow d={\left(-18\right)}^2-4*\left(-3\right)*\left(-26\right)=324-312=12.$ .

Here $D>0$ .

Therefore the graph will lie above x axis.

At $x=0$ the $y=-3x^2-18x-26$ will yield $y=-26$ .

Therefore graph will cut y axis at $\left(0,-26\right)$ .

Therefore the graph will be as mentioned below.

  

b

To determine

To draw: the table for the equation $y=-3x^2-18x-26$ .

Answer to Problem 3E

The table of the equation is

  $\begin{array}{cc}x& y\\ -6& -26\\ -5& -11\\ -4& -2\\ -3& 1\\ -2& -2\\ -1& -11\\ 0& -26\end{array}$ .

Explanation of Solution

Given information:

The equation is $y=-3x^2-18x-26$ and the value of $x$ is $-6,-5,-4,-3,-2,-1,0.$

Formula:

We will substitute the value of $x$ in the equation $y=-3x^2-18x-26$ .

Calculation:

Consider the equation $y=-3x^2-18x-26$ .

Since the value of $x$ is $-6,-5,-4,-3,-2,-1,0.$

Recall that we will substitute the value of $x$ in the equation $y=-3x^2-18x-26$ .

Therefore

At $x=-6\Rightarrow y=-3\left(-6*-6\right)-18\left(-6\right)-26=-26$ .

At $x=-5\Rightarrow y=-3\left(-5*-5\right)-18\left(-5\right)-26=-11$ .

At $x=-4\Rightarrow y=-3\left(-4*-4\right)-18\left(-4\right)-26=-2$ .

At $x=-3\Rightarrow y=-3\left(-3*-3\right)-18\left(-3\right)-26=1$ .

At $x=-2\Rightarrow y=-3\left(-2*-2\right)-18\left(-2\right)-26=-2$ .

At $x=-1\Rightarrow y=-3\left(-1*-1\right)-18\left(-1\right)-26=-11$ .

At $x=0\Rightarrow y=-3\left(0*0\right)-18\left(0\right)-26=-26.$

Therefore the table will be as mentioned below.

  $\begin{array}{cc}x& y\\ -6& -26\\ -5& -11\\ -4& -2\\ -3& 1\\ -2& -2\\ -1& -11\\ 0& -26\end{array}$