Quadratic functions have special features that help us understand their graphs. They can be written in the standard form:

$y = ax^2 + bx + c$

In this equation, (a), (b), and (c) are numbers, and (a) cannot be zero. The number (a) tells us which way the graph will open.

• If (a) is greater than zero (a positive number), the graph opens up.

• If (a) is less than zero (a negative number), the graph opens down.

Here are some important parts of quadratic functions:

• Vertex: This is the highest or lowest point of the graph, called the parabola. We can find the x-coordinate of the vertex using this formula:

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

To get the y-coordinate, we plug this x value back into the equation.

• Axis of Symmetry: This is a vertical line that goes right through the vertex. We can find it using the same formula for the x-coordinate of the vertex:

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

This line divides the parabola into two mirrored halves.

• Y-intercept: This is where the graph crosses the y-axis. We find it using the (c) in the equation. The point will be ((0, c)).

• X-intercepts (Roots): These are the points where the graph crosses the x-axis. We find these points by solving the equation:

$ax^2 + bx + c = 0$

We can also use the quadratic formula to find them:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Direction and Width: The value of (a) also affects how wide or narrow the parabola looks. Bigger values of (|a|) make the parabola narrower, while smaller values make it wider.

Having a good understanding of these features helps students easily analyze and graph quadratic functions.