Quadratic functions and equations are important topics in Algebra I. They have a unique U-shaped curve on a graph. The standard way to write a quadratic equation is like this:

y = ax² + bx + c

Here, a, b, and c are numbers, and a cannot be zero.

One key part of quadratic functions is the vertex. This is the highest or lowest point on the U-shape, depending on the value of a.

• If a is greater than zero (a > 0), the U opens up.
• If a is less than zero (a < 0), the U opens down.

You can find the vertex using this formula:

x = -b / (2a)

Another important feature is the axis of symmetry. This is a vertical line that goes through the vertex. You can find it with the same formula:

x = -b / (2a)

This line splits the U into two identical halves.

Quadratic functions also have intercepts. The y-intercept is where the graph crosses the y-axis. To find it, you plug in 0 for x. This gives you the point (0, c).

The x-intercepts, or roots, are found by solving this equation:

ax² + bx + c = 0

You can find these roots using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Another helpful tool is the discriminant, which is:

D = b² - 4ac

The discriminant tells us about the roots:

• If D > 0, there are two different real roots.
• If D = 0, there is one real root.
• If D < 0, there are no real roots.

Knowing these features helps when you want to graph quadratic functions or solve related equations.