55 Points 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 B: x = -5/2, and x = 1/2,

Part C: f(x)→ + ∞, as x→ − ∞, f(x)→ + ∞, as x→ + ∞ ,

Part D: I would start with determining the vertex. Knowing it's properties ( f(x)→ + ∞, as x→ − ∞ etc. ) we would determine the type of graph, and provided it's x - intercepts we can plot two points on either side and connect them to this vertex and extend beyond infinitely

Step-by-step explanation:

Part B: It would be easier to factor the expression 4x^2 + 8x - 5, firstly;

( 4x^2 - 2x ) + ( 10x - 5 ),

2x ( 2x - 1 ) + 5 ( 2x - 1 ),

Factored Expression: ( 2x + 5 )( 2x - 1 ) ⇒ f ( x ) = ( 2x + 5 )( 2x - 1 )

Now this equation isn't in standard form, but to determine the x - intercept the y - value must still be 0;

0 = ( 2x + 5 )( 2x - 1 ),

2x + 5 = 0, and 2x - 1 = 0,

x = -5/2, and x = 1/2

Part C: Knowing that this graph extends upwards ( an upward facing parabola ), the end behavior would be such;

f(x)→ + ∞, as x→ − ∞,          ⇒        demonstrated by the above

f(x)→ + ∞, as x→ + ∞        

Part D: I would start with determining the vertex. Knowing it's properties ( f(x)→ + ∞, as x→ − ∞ etc. ) we would determine the type of graph, and provided it's x - intercepts we can plot two points on either side and connect them to this vertex and extend beyond infinitely

the formula of a slope:

from a table we have

substitute:

the slope of f(x) is -1.

the graph of funcfion g(x) is a horizontal line. the slope of a horizontal line is equal 0. therefore the slope of g(x) is 0.

-1 < 0