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

see below

Step-by-step explanation:

f(x) = 5x^2 + 2x − 3

Factor

f(x) = (x + 1) (5 x - 3)

To find the x intercepts, set f(x) =0

0 = (x+1) ( 5x-3)

Using the zero product property

x+1 = 0  5x-3 =0

x=-1    5x=3

          x = 3/5

The x intercepts are (-1,0) and (3/5,0)

When x goes to -∞ the 5x^2 term dominates

f( -∞) = 5 (  -∞) ^2 = 5 ( ∞) =  ∞

As x  goes to  -∞     f(x) goes to  ∞

When x goes to ∞ the 5x^2 term dominates

f( ∞) = 5 (  ∞) ^2 = 5 ( ∞) =  ∞

As x  goes to  ∞     f(x) goes to  ∞

We need the end behavior, the x intercepts and the vertex to graph

We know this opens upward since the coefficient of x^2 is positive

The vertex is 1/2 way between the zeros

(-1+3/5) /2 = (-2/5) /2 = -1/5

Substitute this into f(x) to find the minimum

f(-1/5) = (-1/5 + 1) (5 *-1/5 - 3) = 4/5 * (-4) = -16/5

Part A

we multiply by 5

  and we search for two numbers that added give you 2 and multiplied 15, and those are 5 and -3, so that gives us

but because we multiplied by 5, we have to divide by 5

Part B.

Well we know that f(x) intercepts x when f(x)=0

so we have

so we have two cases

when and when

so we solve for them and get

Part C.

well as I understand the end behavior is that in this function is that as x grows towards infinity f(x) also grows towars infinity because the function has a positive leading coefficient.

Part D.

Well we alredy know two points when f(x)=0.

we can also work out when x=0

To find the other points we need to find the vertex.

so the vertex is in x=-0.2

the vertex is in (-0.2,-3.2), and we can graph additional points by giving value to x

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be

the answer is c hope this