Linear Function Study Guide

Linear functions are a fundamental concept in mathematics. They are functions that can be represented by a straight line when graphed on a coordinate plane. The general form of a linear function is given by:

f(x) = mx + b

m is the slope of the line, which represents the rate of change or the steepness of the line.
b is the y-intercept, which is the point where the line crosses the y-axis.

Key Concepts:

Slope: The slope of a linear function measures the rate at which the function is changing. It is calculated as the change in y-values divided by the change in x-values between any two points on the line.
Y-Intercept: The y-intercept is the point where the line crosses the y-axis, and it is represented by the value of b in the function.
Graphing Linear Functions: To graph a linear function, plot the y-intercept first, and then use the slope to find additional points to draw the line.
Equation of a Line: The equation of a straight line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
Linear Equations: Linear functions can also be represented as equations, such as y = 3x + 2.

Practice Problems:

Find the slope and y-intercept of the linear function: f(x) = 2x + 5

Solution:

Slope (m) = 2

Y-Intercept (b) = 5

Graph the linear function: f(x) = -3x + 4

Solution:

Plot the y-intercept at (0, 4), then use the slope to find additional points and draw the line.

Write the equation of the line with a slope of 1/2 and a y-intercept of 3.

Solution:

The equation is y = (1/2)x + 3.

Real-World Applications:

Linear functions are used to model many real-world situations, such as calculating costs, predicting growth rates, and analyzing trends. Understanding linear functions is crucial in fields like economics, physics, engineering, and more.

Overall, mastering the concept of linear functions is essential for developing strong foundational skills in mathematics and its applications.

Good luck with your studies!

◂Math Worksheets and Study Guides Sixth Grade. Multiply Fractions

Create And Print more Fractions worksheets with Comparing Fractions, Equivalent Fractions, and Simplifying Fractions

The resources above cover the following skills:

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Work flexibly with fractions, decimals, and percents to solve problems.

Understand meanings of operations and how they relate to one another.

Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.

Compute fluently and make reasonable estimates.

Select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods.

Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use.

Grade 6 Curriculum Focal Points (NCTM)

Number and Operations: Developing an understanding of and fluency with multiplication and division of fractions and decimals

Students use the meanings of fractions, multiplication and division, and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions and explain why they work. They use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain the procedures for multiplying and dividing decimals. Students use common procedures to multiply and divide fractions and decimals efficiently and accurately. They multiply and divide fractions and decimals to solve problems, including multi-step problems and problems involving measurement.