Finding the minimum or maximum value of a quadratic function is pretty simple once you understand how it works! Quadratic functions look like this:

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

In this formula, $a$, $b$, and $c$ are just numbers we use in our function. The important thing to remember is how the graph looks. This graph, called a parabola, will either smile (open up) if $a$ is positive or frown (open down) if $a$ is negative.

How to Find the Minimum or Maximum:

1. Look at the Value of $a$:

   • If $a > 0, the function has a minimum value (the lowest point).
   • If $a < 0, the function has a maximum value (the highest point).

2. Find the Vertex:

   • The vertex is the tip of the parabola. You can find the $x$-coordinate of the vertex using this formula: $x = -\frac{b}{2a}$

3. Find the $y$-coordinate:

   • Once you have the $x$ value, put it back into the function $f(x)$ to get the $y$ value. This means you'll calculate: $f\left(-\frac{b}{2a}\right)$

4. Understand Your Result:

   • If you were looking for the minimum, the $y$ value you found is your minimum. If you were looking for the maximum, then that value is your maximum!

Example:

Let’s look at the quadratic function $f(x) = -2x^2 + 4x + 1$:

• Here, $a = -2$, so the graph opens down (it frowns).
• Calculate $x = -\frac{4}{2 \cdot -2} = 1$.
• Now, plug $x = 1$ back into the function to find $f(1) = 3$. So, the maximum value is 3!

And that’s how you find the highest and lowest points of quadratic functions! It all comes down to that vertex!