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Parabolas

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Parabolas

Study Guide

Key Definition

A parabola is the graph of a quadratic function of the form $f(x) = ax^2 + bx + c$ where $a \neq 0$.

Important Notes

The vertex of a parabola is the highest or lowest point depending on its orientation.
If $a > 0$, the parabola opens upward.
If $a < 0$, the parabola opens downward.
The axis of symmetry of a parabola is given by $x = -\frac{b}{2a}$.
The y-intercept of the parabola is $f(0) = c$.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$x^2$ denotes x squared

$\frac{a}{b}$ represents a divided by b

$\sqrt{x}$ represents the square root of x

Remember to use proper notation when solving problems

Why It Works

Parabolas are derived from quadratic functions, which are polynomial equations of degree 2. Their shape is determined by the coefficient $a$.

Remember

The vertex form of a parabola is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex.

Quick Reference

Vertex Formula:$x = -\frac{b}{2a}$

Direction of Opening:$a > 0$ (upward), $a < 0$ (downward)

Understanding Parabolas

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A simple explanation with $f(x) = ax^2 + bx + c$ and how it forms a U-shaped curve.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the vertex of the parabola $y = x^2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you're launching a water balloon, and its path is modeled by $y = -2x^2 + 4x + 3$. Determine the maximum height it reaches.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What happens to the parabola $y = ax^2$ as $a$ increases?

Challenge Quiz

Single Choice Quiz

Advanced

For the quadratic function $f(x) = 3x^2 - 6x + 2$, what is the axis of symmetry?

Recap

Watch & Learn

Review key concepts and takeaways