a. Graph of \(y=f(x)\):

Solution:

1. Intercepts:

y-intercept: \(f(0) = -1\) → \((0, -1)\)

x-intercept: \(4x+2=0\) → \((-0.5, 0)\)

2. Asymptotes:

Vertical asymptote: \(x=2\)

Horizontal asymptote: \(y=4\)

b. Range of \(f\):

Solution:

The function never attains the value \(y=4\) (horizontal asymptote)

Range: \(y \in (-\infty, 4) \cup (4, \infty)\) or \(y \neq 4\)

c. Showing \(p=\frac{9}{2}\):

Solution:

Using sum of roots: \(-\frac{1}{2} + p = -\frac{b}{a} = 4\) (since axis of symmetry \(x=2\) implies \(b=-4\) when \(a=1\))

Therefore: \(p = 4 + \frac{1}{2} = \frac{9}{2}\)

d. Values of \(b\) and \(c\):

Solution:

From axis of symmetry: \(-\frac{b}{2} = 2\) ⇒ \(b = -4\)

Using product of roots: \(-\frac{1}{2} \times \frac{9}{2} = c\) ⇒ \(c = -\frac{9}{4}\)

e. Vertex y-coordinate:

Solution:

Substitute \(x=2\) into \(g(x)\):

\(y = (2)^2 -4(2) -\frac{9}{4} = -\frac{25}{4}\)

f. Product of solutions to \(f(x)=g(x)\):

Solution:

Solve \(\frac{4x+2}{x-2} = x^2 -4x -\frac{9}{4}\)

Multiply through by \(x-2\) and rearrange:

\(x^3 -6x^2 + \frac{7}{4}x + \frac{1}{2} = 0\)