LINEAR FUNCTION

Linear Function

Definition Of Linear Function

A function that can be graphically represented in the Cartesian coordinate plane by a straight line is called a Linear Function.

More About Linear Function

A linear function is a first degree polynomial of the form, F(x) = m x + c, where m and c are constants and x is a real variable.
The constant m is called slope and c is called y-intercept.

Example of Linear Function

y = 3x + 5 is a linear function.
The graph of the function y = 2x is shown below. This is a linear function since the points fit onto a straight line

example of Linear Function

Video Examples: Graphing Linear Functions by Finding X,Y Intercept

Solved Example on Linear Function

Ques: Identify the graph that represents a linear function.

example of Linear Function

Choices:

A. Graph 1
B. Graph 3
C. Graph 4
D. Graph 2

Correct Answer: C

Solution:

Step 1: The graph of a linear function is a straight line.
Step 2: Graph 4 is a straight line.
Step 3: So, Graph 4 represents a linear function.

Quick Summary

A linear function is a first-degree polynomial.
It can be expressed in the form F(x) = mx + c, where m is the slope and c is the y-intercept.
The graph of a linear function is a straight line.

\[ F(x) = mx + c \]

⚠️ Common Mistakes

Confusing slope and y-intercept.
Assuming all functions are linear.
Incorrectly plotting points on the coordinate plane.

🌍 Real-World Uses

Calculating the cost of a taxi ride, where the initial fee is 'c' and the cost per mile is 'm'.
Modeling the depreciation of an asset over time, assuming a constant rate of depreciation.
Converting temperature from Celsius to Fahrenheit using the formula F = (9/5)C + 32.

📋 Standards Alignment

CCSS.MATH.HSF.LE.A.1
CCSS.MATH.HSF.IF.C.7

🍎 Teacher Insights

Use real-world examples to illustrate linear functions, such as distance vs. time or cost vs. quantity. Emphasize the relationship between the equation, graph, and table of values.

🎓 Prerequisites

Basic Algebra
Coordinate Plane
Slope

Check Your Knowledge

Q1: Which of the following equations represents a linear function?

A. y = x^2 + 1 B. y = 2x + 3 C. y = 1/x D. y = sqrt(x)

Q2: What is the slope of the linear function y = 5x - 2?

A. -2 B. 2 C. 5 D. -5

Frequently Asked Questions

Q: What is the slope of a linear function?
A: The slope (m) represents the rate of change of the function. It indicates how much the y-value changes for every unit change in the x-value.

Q: What is the y-intercept of a linear function?
A: The y-intercept (c) is the point where the line crosses the y-axis. It is the value of y when x is 0.