Question

HELP ASAP NEED HELP. WILL GIVE BRAINIEST AND 100 POINTS.

Answer 1

9514 1404 393

Answer:

  a.  f(x) = (4x +5)(x -3)

  b.  x-intercepts: {-1.25, +3}

  c.  f(x) goes to +infinity as x goes to ±infinity

  d. plot the x-intercepts, the vertex, and a few other points; draw a smooth curve through them

Step-by-step explanation:

a. Here, you're looking for factors of (4)(-15) = -60 that have a sum of -7. Those factors will be -12 and +5, so we can factor the function as ...

  f(x) = (4x -12)(4x +5)/4

  f(x) = (x -3)(4x +5)

__

b. The x-intercepts are the values of x that make the factors zero:

  x -3 = 0   ⇒   x = 3 . . . . (add 3)

  4x +5 = 0   ⇒   x = -5/4 . . . . (add -5, divide by 4)

__

c. As with all even-degree polynomials with positive leading coefficients, the end behavior of f(x) matches that of |x|:

  as x → ∞, f(x) → ∞

  as x → -∞, f(x) → ∞

__

d. The vertex of the function will be located halfway between the x-intercepts, at x = (3-5/4)/2 = (7/4)/2 = 7/8. The value of f(x) there is ...

  f(7/8) = (4(7/8) +5)(7/8 -3) = -289/16 = -18.0625

Since the leading coefficient is 4, the normal x^2 function behavior is stretched vertically by a factor of 4. So, additional points would be

  x = 7/8 ±1, f(x) = -18.0625 +4·1 = -14.0625

  x = 7/8 ±2, f(x) = -18.0625 +4·2^2 = -2.0625

The answers of part B locate two points on the graph, and help find the vertex. The answer of part C confirms the graph as opening upward.

__

Additional comment

I find a graphing calculator to be very helpful finding the roots and making the graph.