Converting between the standard form and vertex form of quadratic functions might seem a little confusing at first. But don’t worry! Once you learn the steps, it’s really not too hard. Let’s go through it step by step.

Standard Form

The standard form of a quadratic function looks like this:

$f(x) = ax^2 + bx + c$

In this equation, $a$, $b$, and $c$ are just numbers. This form is great for figuring out where the graph touches the y-axis, also known as the y-intercept. This point is found at $(0, c)$ on the graph.

Vertex Form

Now let’s talk about the vertex form. It looks like this:

$f(x) = a(x - h)^2 + k$

In this case, $(h, k)$ is the vertex of the graph. The vertex is the highest or lowest point, depending on whether the parabola opens up or down. This form is super handy when you want to quickly find the vertex or see how the graph changes.

Converting from Standard to Vertex Form

1. Start with Standard Form: Begin with $f(x) = ax^2 + bx + c$ and turn it into vertex form.

2. Complete the Square: This is the key step.

   • First, if $a$ is not 1, you will need to take it out of the $x^2$ and $bx$ terms.
   • Next, take half of the $b$ value, square it, and add and subtract that number inside the parentheses.
   • Here’s how it looks:
   $$f(x) = a(x^2 + \frac{b}{a}x) + c$$

   Then, you’ll add and subtract $\left(\frac{b}{2a}\right)^2$. It will start looking like this:

   $$f(x) = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c$$
   • After simplifying, you’ll have it in vertex form!

Converting from Vertex to Standard Form

1. Start with Vertex Form: Now you have $f(x) = a(x - h)^2 + k$.

2. Expand the Square: First, expand $(x - h)^2$:

   $$(x - h)^2 = x^2 - 2hx + h^2$$
   • Then, multiply the $a$ through and combine similar terms:
   $$f(x) = ax^2 - 2ahx + (ah^2 + k)$$

And just like that, you have the quadratic in standard form!

With practice, these conversions will become much easier. Don't forget to try out a few problems on your own to help understand it better. Happy graphing!