A

use the function f(x) = 4x2 + 8x − 5 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)

Part A)

Part B) The x-intercepts are the points (-2.5,0) and (0.5,0)

Part C) see the explanation

Part D) The graph in the attached figure

Step-by-step explanation:

we have

Part A: Completely factor

Factor 4 leading coefficient

Complete the square

Rewrite as perfect squares

Part B: What are the x-intercepts of the graph of f(x)?

we know that

The x-intercepts are the values of x when the value of the function is equal to zero

we have

For f(x)=0

Move the constant to the right side

take the square root both sides

The x-intercepts are the points (-2.5,0) and (0.5,0)

Part C: Describe the end behavior of the graph of f(x)

we have

This is the equation of a vertical parabola in vertex form

The parabola open upward

The vertex represent a minimum

The vertex is the point (-1,-9)

The function is decreasing in the interval (-∞,-1)

The function is increasing in the interval (-1,∞)

The domain is the interval (-∞,∞)

The range is the interval [-9,∞)

Part D: What are the steps you would use to graph f(x)?

step 1

Plot the vertex

The vertex is the point (-1,-9)

step 2

Plot the y-intercept

The y-intercept is the value of the function when the value of x is equal to zero

For x=0

The y-intercept is the point (0,-5)

step 3  

Plot the x-intercepts    

The x-intercepts are the points (-2.5,0) and (0.5,0)

using a graphing tool

join the points to graph the parabola

see the attached figure

x =-.20752

step-by-step explanation:

4^(x+2)=12

4^(x+2)=12

take the log base 4 on each side

log 4 (4^(x+2) = log 4(12)

we know log b(a^y) = y log b (a)

(x+2) log 4(4) = log 4 (12)

x+2 = log 4 (12)

we know   logb c = loga c/loga b

we want to convert to base 10   so a = 10

by default, we do not write the 10   c = 4   and b = 10

log4 10 = log 12/log 4

log4 10 = log 12/ log 4

x+2 = log 12/ log 4

subtract 2 from each side

x +2 -2 = log 12/ log 4 -2

x = 1.79248125 -2

x =-.20752