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Graphing Quadratic Equations using the Axis of Symmetry

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Graphing Quadratic Equations using the Axis of Symmetry

Study Guide

Key Definition

A quadratic equation is in the form $y = ax^2 + bx + c$, where $a \neq 0$. The axis of symmetry is $x = \frac{-b}{2a}$.

Important Notes

The vertex of the parabola is at $\left( \frac{-b}{2a}, f\bigl(\frac{-b}{2a}\bigr) \right)$, where the y-coordinate is found by substituting x = \frac{-b}{2a} into the quadratic
The axis of symmetry is a vertical line
The parabola is symmetric with respect to this line
The x-coordinate of the vertex is $\frac{-b}{2a}$
Substitute x-values to find corresponding y-values

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$\frac{a}{b}$ represents a fraction

Remember to use proper notation when solving problems

Why It Works

The formula $x = \frac{-b}{2a}$ gives the x-value of the vertex, which is the highest or lowest point of the parabola.

Remember

The axis of symmetry helps in plotting parabolas symmetrically.

Quick Reference

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

Vertex:$\left( \frac{-b}{2a}, y \right)$

Understanding Graphing Quadratic Equations using the Axis of Symmetry

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The axis of symmetry is the line $x = \frac{-b}{2a}$. It divides the parabola into two mirror images.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the axis of symmetry for $y = x^2 - 7x + 2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder jumps off a ramp, and the path is modeled by $y = -x^2 + 6x + 1$. Find the highest point of the jump.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

A parabola opens upwards. If the axis of symmetry is $x = 4$, what can you deduce about the coefficients of the quadratic equation?

Challenge Quiz

Single Choice Quiz

Advanced

Find the vertex of the parabola $y = 2x^2 - 8x + 3$.

Recap

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Review key concepts and takeaways