#Quadratic Functions: Your Ultimate Guide 🚀

Hey there! Quadratic graphs might seem intimidating, but they're actually super friendly once you know their secrets. Think of them as the superheroes of the math world, modeling everything from a ball's flight to a company's profits. This guide is your cheat sheet for mastering them on the AP SAT (Digital) exam. Let's dive in!

#Understanding Quadratic Functions: The Basics

#Definition and Standard Form

• A quadratic function is a polynomial of degree 2. Think of it as a fancy way of saying it has an $$x^2$$ term.
• It's written in standard form as: $$f(x) = ax^2 + bx + c$$, where a, b, and c are just numbers, and a can't be zero. If a was zero, it wouldn't be quadratic anymore!
• The graph of a quadratic function is a smooth, U-shaped curve called a parabola. It's always symmetrical.

#Key Points on the Graph

• Y-intercept: Where the graph crosses the y-axis. Just plug in $$x = 0$$ into the function, so it's $$f(0)$$. Easy peasy!
• X-intercepts (roots): Where the graph crosses the x-axis. These are the solutions to the equation $$ax^2 + bx + c = 0$$. You'll often use factoring, the quadratic formula, or completing the square to find these.
• Vertex: The highest or lowest point of the parabola. It's like the peak or the valley of the U-shape. It always sits right on the axis of symmetry.

#Axis of Symmetry, Vertex, and Opening

#Axis of Symmetry

• It's a vertical line that cuts the parabola perfectly in half. It's like the mirror line for the U-shape.
• The equation for the axis of symmetry is: $$x = -b/(2a)$$. Remember this, it's a lifesaver!
• a and b are the coefficients from the standard form of the quadratic equation.

#Vertex Formula and Significance

• The vertex is found using the formula: $$(-b/(2a), f(-b/(2a)))$$. The x-coordinate is the same as the axis of symmetry, and the y-coordinate is what you get when you plug that x-value back into the function.
• It's the maximum or minimum point of the parabola. If the parabola opens upwards, it's a minimum; if it opens downwards, it's a maximum.

#Direction of Openin...