Question

Using the function f(x)=5x^2+2x-3 to answer the following questions.1. Completely factor f(x).2. What are the d-intercepts of the graph of f(x)? Show work.3. Describe the end behavior of the graph of f(x). Explain.4. What are the steps you would use to graph f(x)? Justify that you can use answers obtained in Part B and Part C to draw the graph.

Answer 1

The function is given below as

`f(x)=5x^2+2x-3`

Step 1:

Completely factor f(x)

Here will find two factors that we will multiply to give -15(5*-3)

The same two numbers will be added to give (+2)

The two number are +5 and -3

Hence,

We will replace +2x with +5x-3x

`\begin{gathered} f(x)=5x^2+2x-3 \\ f(x)=5x^2+5x-3x-3 \\ f(x)=5x(x+1)-3(x+1) \\ f(x)=(5x-3)(x+1) \end{gathered}`

Hence,

By completely factoring, we will have it be

`\Rightarrow f(x)=(5x-3)(x+1)`

PART 2:

What are the d-intercepts of the graph of f(x)

To figure out the y-intercepts, we will put the value of x=0

`\begin{gathered} \begin{equation*} f(x)=(5x-3)(x+1) \end{equation*} \\ f(0)=(5(0)-3)(0+1) \\ f(0)=-3 \end{gathered}`

Hence,

The y-intercept is

`\Rightarrow(0,-3)`

To figure out the x-intercept,

we will put the value of y=0

`\begin{gathered} \begin{equation*} f(x)=(5x-3)(x+1) \end{equation*} \\ y=f(x)=(5x-3)(x+1) \\ (5x-3)(x+1)=0 \\ 5x-3=0,x+1=0 \\ 5x=3,x=-1 \\ (5x)/(5)=(3)/(5),x=-1 \\ x=(3)/(5),x=-1 \end{gathered}`

Hence,

The x-intercepts are

`\Rightarrow(-1,0)\text{ }and\text{ }((3)/(5),0)`

Part 3:

Describe the end behavior of the graph of f(x)

What is the End Behavior of a Function? The end behavior of a function f(x) refers to how the function behaves when the variable x increases or decreases without bound. In other words, the end behavior describes the ultimate trend in the graph of f(x) as we move towards the far right or far left of the x -axis.

Hence,

The end behavior of the function f(x) is given below as

`\begin{gathered} \Rightarrow\mathrm{as}\:x\to\:-\infty\:,\:f\left(x\right)\to\:+\infty\: \\ \Rightarrow\mathrm{as}\:x\to\:+\infty\:,\:f\left(x\right)\to\:+\infty \end{gathered}`

As x approaches negative infinity, f(x) approaches positive infinity and as x approaches positive infinity, f(x) approaches positive infinity

Part 4:

Steps to plot the graph of f(x)

We will calculate the y-intercepts and x-intercepts as we did in part 2

`\begin{gathered} y-intercept=(0,-3) \\ x-intercept=(-1,0),(0.6,0) \end{gathered}`

We will know the end behavior and it opens upwards because the leading coefficient is positive

We will calculate the vertex of the parabola

`\begin{gathered} (h,k) \\ h=-(b)/(2a),a=5,b=2,c=-3 \\ h=-(2)/(2(5))=-(1)/(5) \\ k=5x^2+2x-3 \\ k=5((1)/(5))^2+2(-(1)/(5))-3 \\ k=(1)/(5)-(2)/(5)-3 \\ k=(-1)/(5)-3 \\ k=(-1-15)/(5) \\ k=-(16)/(5)=-3.2 \\ (-0.2,-3.2) \end{gathered}`

By graphing this parabola, we will have the graph be