Questions: g(x)=(x^2+x-20)/(x+3) Hole(s) VA x=-3 Zeros Y-int -5 EBA y=-5

Solution Steps

To solve the given problem, we need to analyze the rational function \( g(x) = \frac{x^2 + x - 20}{x + 3} \). We will:

1. Determine if the graph crosses the EBA (End Behavior Asymptote).
2. Identify any holes in the graph.
3. Confirm the vertical asymptote (VA) at \( x = -3 \).
4. Find the zeros of the function.
5. Verify the y-intercept.

1. Crossing the EBA: To check if the graph crosses the EBA, we set \( g(x) \) equal to the EBA and solve for \( x \).
2. Holes: Holes occur where both the numerator and denominator are zero. We factor the numerator and check for common factors with the denominator.
3. Vertical Asymptote (VA): The VA is given as \( x = -3 \). We confirm this by checking where the denominator is zero.
4. Zeros: Zeros are found by setting the numerator equal to zero and solving for \( x \).
5. Y-intercept: The y-intercept is found by evaluating \( g(0) \).

To check if the graph of \( g(x) = \frac{x^2 + x - 20}{x + 3} \) crosses the EBA \( y = -5 \), we solve: \[ g(x) = -5 \] \[ \frac{x^2 + x - 20}{x + 3} = -5 \] \[ x^2 + x - 20 = -5(x + 3) \] \[ x^2 + x - 20 = -5x - 15 \] \[ x^2 + 6x - 5 = 0 \] Solving this quadratic equation, we get: \[ x = -3 + \sqrt{14} \quad \text{and} \quad x = -3 - \sqrt{14} \]

Holes occur where both the numerator and denominator are zero. The numerator is \( x^2 + x - 20 \) and the denominator is \( x + 3 \). Factoring the numerator: \[ x^2 + x - 20 = (x - 4)(x + 5) \] Since there are no common factors with the denominator \( x + 3 \), there are no holes.

The vertical asymptote occurs where the denominator is zero: \[ x + 3 = 0 \] \[ x = -3 \]

Zeros are found by setting the numerator equal to zero: \[ x^2 + x - 20 = 0 \] \[ (x - 4)(x + 5) = 0 \] Thus, the zeros are: \[ x = 4 \quad \text{and} \quad x = -5 \]

The y-intercept is found by evaluating \( g(0) \): \[ g(0) = \frac{0^2 + 0 - 20}{0 + 3} = \frac{-20}{3} \]

Final Answer