What is the axis of symmetry and vertex for the graph  y=x^2 + 3x - 4?

Nov 22, 2017

The vertex is $\left(- \frac{3}{2} , - \frac{25}{4}\right)$ and the line of symmetry is $x = - \frac{3}{2}$.

Explanation:

$y = {x}^{2} + 3 x - 4$

There are a couple ways to find the vertex - using $- \frac{b}{2 a}$ or converting it to vertex form. I'll show doing it both ways.

Method 1 (probably better method): $x = - \frac{b}{2 a}$

The equation is in standard quadratic form, or $a {x}^{2} + b x + c$.
Here, $a = 1$, $b = 3$, and $c = - 4$.

To find the x-coordinate of the vertex in standard form, we use $- \frac{b}{2 a}$. So...
${x}_{v} = - \frac{3}{2 \left(1\right)}$
${x}_{v} = - \frac{3}{2}$

Now, to find the y-coordinate of the vertex, we plug in our x-coordinate of the vertex back into the equation:
$y = {\left(- \frac{3}{2}\right)}^{2} + 3 \left(- \frac{3}{2}\right) - 4$
$y = \frac{9}{4} - \frac{9}{2} - 4$
$y = \frac{9}{4} - \frac{18}{4} - \frac{16}{4}$
$y = - \frac{25}{4}$

So our vertex is $\left(- \frac{3}{2} , - \frac{25}{4}\right)$.

If you think about it, the axis of symmetry is the line of the x-coordinate because that is where there is a 'reflection' or where it becomes symmetrical.
So this means that the line of symmetry is $x = - \frac{3}{2}$

Method 2: Converting into vertex form
We can also convert this equation into vertex form by factoring. We know that the equation is $y = {x}^{2} + 3 x - 4$.

To factor this, we need to find 2 numbers that multiply up to -4 AND add up to 3. $4$ and $- 1$ work because $4 \cdot - 1 = - 4$ and $4 - 1 = 3$.
So it's factored into $\left(x + 4\right) \left(x - 1\right)$

Now our equation is $y = \left(x + 4\right) \left(x - 1\right)$ which is in vertex form.

First, we need to find the x-intercepts (what x is when y = 0). To do this, let's set:
$x + 4 = 0$ and $x - 1 = 0$
$x = - 4$ and $x = 1$.

To find the x-coordinate of the vertex, we find the average of the 2 x-intercepts. Average is $\frac{{x}_{1} + {x}_{2}}{2}$
${x}_{v} = \frac{- 4 + 1}{2}$

${x}_{v} = - \frac{3}{2}$
(As you can see, it brings the same result as in $- \frac{b}{2 a}$.)

To find the y-coordinate of the vertex, we would pluck the x-coordinate of the vertex back into the equation and solve for y, just like we did in method 1.
You can watch this video if you still need help solving these: http://virtualnerd.com/algebra-1/quadratic-equations-functions/graphing/graph-basics/vertex-example

Hope this helps (sorry that it's so long)!