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Linear Function

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Linear Function

Study Guide

Key Definition

A linear function is defined as any function that can be written in the form $f(x) = mx + b$ where $m$ and $b$ are real numbers.

Important Notes

The graph of a linear function is a straight line.
The slope $m$ indicates the line's steepness; positive slopes rise, negative slopes fall.
The y-intercept $b$ is where the line crosses the y-axis.
If $m = 0$, it is a constant function.
Linear functions are foundational for understanding more complex mathematical concepts.

Mathematical Notation

$y = mx + b$ is the general form of a linear equation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Linear functions represent constant rates of change, making them predictable and easy to graph using $m$ and $b$.

Remember

The slope-intercept form $y = mx + b$ is key to graphing linear functions.

Quick Reference

Slope Formula:m = \frac{y_2 - y_1}{x_2 - x_1}</math-formula>

Y-Intercept:The point where $x = 0$, $y = b$

Horizontal Line:If $m = 0$, $y = b$

Understanding Linear Function

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A linear function can be visualized as a straight line in the format $f(x) = mx + b$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the slope of the line represented by $y = 2x - 3$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A teenager saves money with a weekly allowance. If they start with $10 and save $5 each week, find the amount saved after $n$ weeks.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If the graph of $y = -3x + 6$ passes through the point $(x, y)$, find $x$ when $y = 0$.

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following lines is perpendicular to $y = \frac{1}{2}x + 4$?

Recap

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Review key concepts and takeaways