The problem asks us to graph the quadratic equation $y = x^2 + 2x - 8$ on the given coordinate plane. We also need to describe similarities and differences between quadratics, parabolas, and other types of graphs we might have encountered.

1. Problem Description

The problem asks us to graph the quadratic equation $y = x^2 + 2x - 8$ on the given coordinate plane. We also need to describe similarities and differences between quadratics, parabolas, and other types of graphs we might have encountered.

2. Solution Steps

To graph the quadratic equation, we can find the vertex, x-intercepts, and y-intercept.

First, let's find the x-coordinate of the vertex using the formula:

$x_v = -\frac{b}{2a}$

In our equation $y = x^2 + 2x - 8$, $a = 1$ and $b = 2$. Thus,

$x_v = -\frac{2}{2(1)} = -1$.

Now, we can find the y-coordinate of the vertex by substituting $x_v = -1$ into the equation:

$y_v = (-1)^2 + 2(-1) - 8 = 1 - 2 - 8 = -9$.

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

Next, let's find the x-intercepts by setting $y = 0$:

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

We can factor this quadratic equation:

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

This gives us two solutions: $x = -4$ and $x = 2$. So, the x-intercepts are $(-4, 0)$ and $(2, 0)$.

Finally, let's find the y-intercept by setting $x = 0$:

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

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

Now, we can plot these points on the graph: vertex $(-1, -9)$, x-intercepts $(-4, 0)$ and $(2, 0)$, and y-intercept $(0, -8)$. We can also find a few other points to help us sketch the graph, such as when $x = -5$, $y = (-5)^2 + 2(-5) - 8 = 25 - 10 - 8 = 7$, and when $x = 3$, $y = (3)^2 + 2(3) - 8 = 9 + 6 - 8 = 7$. So, the points $(-5, 7)$ and $(3, 7)$ are on the graph.

The quadratic equation forms a parabola when graphed. Parabolas are symmetrical around their vertex. Quadratics and parabolas are related because the graph of a quadratic equation is a parabola. They are different from linear equations (which are straight lines) or exponential functions (which have curves that increase or decrease rapidly). Parabolas have a single vertex point where the graph changes direction, and a line of symmetry that goes through the vertex.

3. Final Answer

The graph of $y = x^2 + 2x - 8$ is a parabola with vertex at $(-1, -9)$, x-intercepts at $(-4, 0)$ and $(2, 0)$, and y-intercept at $(0, -8)$. Quadratics and parabolas are similar because a quadratic equation's graph is a parabola. They are different from other graphs like lines or exponentials in terms of their shape and properties, such as symmetry.