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Graphing Quadratic Equations using Transformations

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Graphing Quadratic Equations using Transformations

Study Guide

Key Definition

A quadratic function is of the form $y = ax^2 + bx + c$ where $a \neq 0$. The graph is a parabola.

Important Notes

The graph of $y = x^2 + c$ is a parabola shifted $c$ units up.
The graph of $y = ax^2$ stretches vertically by a factor of $a$ if |a| > 1 and compresses vertically if 0 < |a| < 1.
Negative $a$ flips the parabola upside down.
Horizontal shifts occur with $(x - h)^2$ form.
Vertical shifts occur by adding/subtracting from $c$.
Interactive graph for vertical shifts: https://www.desmos.com/calculator/vertical_shift
Interactive graph for horizontal shifts: https://www.desmos.com/calculator/horizontal_shift
Interactive graph for stretches and compressions: https://www.desmos.com/calculator/stretch_compression
Interactive graph for reflections: https://www.desmos.com/calculator/reflection

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Transformations allow us to understand how changes in the equation affect the graph's shape and position, using $y = ax^2 + bx + c$.

Remember

The vertex form $y = a(x - h)^2 + k$ shows the vertex at $(h, k)$.

Quick Reference

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

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

Understanding Graphing Quadratic Equations using Transformations

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

We start with y = x^2, which is a parabola symmetric about the y-axis with vertex at (0,0). Adding a constant c shifts it up or down. For example, y = x^2 + 3 moves the vertex to (0,3). See interactive example: https://www.desmos.com/calculator/vertical_shift.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What transformation does $y = x^2 + 3$ represent?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder launches off a ramp described by $y = -x^2 + 6x - 8$. How high does the ramp launch the skateboarder?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the transformation $y = -\frac{1}{2}(x - 4)^2 + 5$. What are the transformations?

Challenge Quiz

Single Choice Quiz

Advanced

Given $y = 3(x + 2)^2 - 7$, which describes the transformations?

Recap

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Review key concepts and takeaways