c) identify the following for this equation: $y = x^2 + 2x - 8$\n\n$\\frac{-b}{2a} \\rightarrow$\n\n$y = x^2…

Explanation:

Step1: Identify coefficients from the quadratic equation

For $y = x^2 + 2x - 8$, the coefficients are $a = 1$, $b = 2$, and $c = -8$.

Step2: Calculate the x-coordinate of the vertex

$$x = \frac{-b}{2a} = \frac{-2}{2(1)} = -1$$

Step3: Calculate the y-coordinate of the vertex

$$y = (-1)^2 + 2(-1) - 8 = 1 - 2 - 8 = -9$$

Step4: Determine if it is a minimum or maximum

Since $a = 1 > 0$, the parabola opens upward, indicating a minimum.

Step5: Identify the axis of symmetry

The axis of symmetry is the vertical line through the vertex $x = -1$.

Step6: Find the y-intercept

Set $x = 0$: $y = (0)^2 + 2(0) - 8 = -8$. The point is $(0, -8)$.

Step7: Find the x-intercepts

Set $y = 0$ and factor: $x^2 + 2x - 8 = (x + 4)(x - 2) = 0$. Thus, $x = -4$ and $x = 2$.

Step8: Determine domain and range

The domain for any quadratic is all real numbers. The range starts from the minimum $y$-value.

Answer:

Vertex: $(-1, -9)$ Min or Max?: Min Axis of Symmetry: $x = -1$ y-intercept: $(0, -8)$ x-intercept(s): $(-4, 0)$ and $(2, 0)$ Domain: $(-\infty, \infty)$ Range: $[-9, \infty)$