# How do you identify the important parts of y=x^2-1 to graph it?

Oct 19, 2015

In summary, find the vertex and x-intercepts, and plug and chug for additional points. Finally, connect 'em all together with a neat curve.

#### Explanation:

The "important parts" in terms of graphing would be the x-intercepts and the vertex. From there you can just plug and chug to identify other points on the graph.

To find the x-intercepts, you set $y = 0$ and solve for your $x$s:
$0 = {x}^{2} - 1$
$1 = {x}^{2}$ (Adding 1 to both sides)
$x = \pm 1$ (Taking square roots)

Thus, the x-intercepts occur at $x = 1$ and $x = - 1$.

The vertex is the "beginning" point of a quadratic like this one. In other words, it's where the two curves meet on a parabola (not a very good definition, but it goes). To find the x-coordinate of the vertex, we use the formula $x = - \frac{b}{2 a}$, where $a$ and $b$ are the numbers in $a {x}^{2} + b x + c$ (which is the form of quadratic presented in this problem). In our equation ${x}^{2} - 1$, $a = 1$ and $b = 0$ (since there is no middle term, we have $b = 0$). Thus, $x = - \frac{0}{2 \left(1\right)} = 0$. This is the x-coordinate of the vertex; to find the y, we simple plug in: $y = {x}^{2} - 1 = {\left(0\right)}^{2} - 1 = - 1$. The coordinates of the vertex are $\left(0 , - 1\right)$.

We can see everything above on the graph below. The vertex, $\left(0 , - 1\right)$, is the bottom of the graph; you'll want to start there and work your way up. Next are the intercepts at -1 and 1 on the x-axis; plot those points, and connect them (vertex, and both intercepts) together with a nice curve. Finally, choose x-values like -3, -2, 2, and 3 and find which y-values they produce:
$y = {\left(- 3\right)}^{2} - 1 = 9 - 1 = 8$
$y = {\left(- 2\right)}^{2} - 1 = 4 - 1 = 3$
$y = {\left(2\right)}^{2} - 1 = 4 - 1 = 3$
$y = {\left(3\right)}^{2} - 1 = 9 - 1 = 8$
And just like that, we've identified 4 more points to plot:
$\left(- 3 , 8\right)$
$\left(- 2 , 3\right)$
$\left(2 , 3\right)$
$\left(3 , 8\right)$

Finally finally, to complete the rough sketch of the graph, connect all the points together - the "special" ones (vertex and intercept(s)) and the ones we found by plugging and chugging.

graph{x^2-1 [-10, 10, -5, 5]}