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Graphing Quadratic Equations

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Graphing Quadratic Equations

Study Guide

Key Definition

The standard form of a quadratic equation is $y = ax^2 + bx + c$ where $a \neq 0$.

Important Notes

The y-intercept of a parabola is $(0, c)$.
If $a > 0$, the parabola opens upward.
If $a < 0$, the parabola opens downward.
The vertex form of a quadratic is $y = a(x - h)^2 + k$.
The vertex of the parabola is $(h, k)$ in vertex form.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$x^2$ is x squared

Remember to use proper notation when solving problems

Why It Works

A quadratic function $y = ax^2 + bx + c$ produces a parabola, a symmetric U-shaped curve. By completing the square, we find the vertex at $x = -\tfrac{b}{2a}$ and its y-coordinate $k = f(-\tfrac{b}{2a})$. The coefficient $a$ controls the direction (upward if $a>0$, downward if $a<0$) and the width of the parabola. The axis of symmetry is the vertical line through the vertex, and the quadratic formula gives the roots where the parabola crosses the x-axis. Understanding these transformations explains why graphing follows from the algebraic form.

Remember

The vertex form $y = a(x - h)^2 + k$ makes it easy to identify the vertex of the parabola.

Quick Reference

Standard Form:$y = ax^2 + bx + c$

Vertex Form:$y = a(x-h)^2 + k$

Axis of Symmetry:$x = -\frac{b}{2a}$

Understanding Graphing Quadratic Equations

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Video explanation of this concept

Beginner

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Beginner Explanation

A quadratic function can be written as $y = ax^2 + bx + c$. Its graph is a U-shaped curve called a parabola. For example, $y = x^2$ opens upward with vertex at the origin. The constant term $c$ shifts the graph up or down. (See the diagram of $y = x^2$.)

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the y-intercept of the equation $y = 2x^2 - 3x + 5$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are designing a skateboard ramp shaped like a parabola. The equation is $y = -0.5x^2 + 2x + 3$. Find the maximum height.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the quadratic equation $y = x^2 - 4x + 4$. How would the graph change if the equation becomes $y = x^2 - 4x + 8$?

Challenge Quiz

Single Choice Quiz

Advanced

For the equation $y = 3x^2 - 12x + 9$, what is the vertex?

Recap

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Review key concepts and takeaways