Quadratic functions are a key part of algebra. They have their own special form, and we can show them with graphs. Let’s break down the important parts:

1. Standard Form

A quadratic function is usually written like this:

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

Here:

• $a$, $b$, and $c$ are numbers that stay the same.
• The number $a$ tells us if the graph goes up or down:
  • If $a$ is more than 0, the graph opens upwards.
  • If $a$ is less than 0, the graph opens downwards.

2. Vertex

The vertex is the highest or lowest point on the graph. You can find it using this formula:

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

Once you find $x$, you can put it back into the function to get the $y$-value of the vertex. This point is important because it shows how the function behaves.

3. Axis of Symmetry

Every quadratic function has a line called the axis of symmetry. This line goes straight up and down through the vertex. It can be found using the same formula:

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

This line splits the graph into two equal parts.

4. Roots or Zeros

Quadratic functions can have:

• Two different roots (which means solutions), if a part of the formula called the discriminant $D = b^2 - 4ac$ is more than 0.
• One root, if $D = 0$.
• No real roots, if $D < 0$.

5. Y-Intercept

The y-intercept is where the graph crosses the y-axis. To find it, we set $x = 0$:

$$f(0) = c$$

6. Graph Shape

The graph of a quadratic function looks like a U shape, called a parabola. How it looks depends on the values of $a$, $b$, and $c$. The steepness and width of the U shape are affected by the value of $a$.

These features help us understand how quadratic functions work in algebra.