The problem asks us to graph the quadratic equation $y = x^2 + 2x - 8$ on a given graph and to discuss the properties of the resulting parabola.

1. Problem Description

The problem asks us to graph the quadratic equation $y = x^2 + 2x - 8$ on a given graph and to discuss the properties of the resulting parabola.

2. Solution Steps

First, we need to find some key points on the parabola to plot. We can find the vertex, the y-intercept, and the x-intercepts.

The x-coordinate of the vertex is given by $x = -b/(2a)$, where $a=1$ and $b=2$.

$x = -2/(2*1) = -1$

To find the y-coordinate of the vertex, we substitute $x = -1$ into the equation:

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

So the vertex is $(-1, -9)$.

To find the y-intercept, we set $x = 0$:

$y = (0)^2 + 2(0) - 8 = -8$

So the y-intercept is $(0, -8)$.

To find the x-intercepts, we set $y = 0$:

$x^2 + 2x - 8 = 0$

We can factor the quadratic equation:

$(x + 4)(x - 2) = 0$

So the x-intercepts are $x = -4$ and $x = 2$. The points are $(-4, 0)$ and $(2, 0)$.

Now we have the vertex $(-1, -9)$, the y-intercept $(0, -8)$, and the x-intercepts $(-4, 0)$ and $(2, 0)$.

We can plot these points and draw the parabola.

Observations about quadratics and parabolas:

- They are U-shaped.

- They have a vertex, which is either a minimum or a maximum point.

- They are symmetrical about the vertical line that passes through the vertex.

- They can have 0, 1, or 2 x-intercepts.

How are they similar?

- Quadratics and parabolas are essentially the same thing. The graph of a quadratic equation is a parabola.

How are they different from other things you've graphed?

- Unlike linear equations (straight lines), parabolas are curved.

- Unlike some other curves, parabolas have a well-defined vertex and symmetry.

3. Final Answer

The graph of $y = x^2 + 2x - 8$ is a parabola with vertex $(-1, -9)$, y-intercept $(0, -8)$, and x-intercepts $(-4, 0)$ and $(2, 0)$. Quadratics and parabolas are the same. Parabolas are different from lines because they are curved. They have a vertex and an axis of symmetry.