Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

3. Functions

Intro to Functions & Their Graphs

College Algebra

3. Functions

Intro to Functions & Their Graphs

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1:37 minutes

Problem 40

Textbook Question

Graph each equation. 3y = x

Verified step by step guidance

Step 1: The given equation is 3y = x. To graph this equation, it's easier to have it in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. So, first, let's rewrite the equation in this form by dividing both sides of the equation by 3 to isolate y. This gives us y = x/3.

Step 2: Now that we have the equation in slope-intercept form, we can identify the slope and the y-intercept. In this case, the slope (m) is 1/3 and the y-intercept (b) is 0, because there is no constant term added or subtracted on the right side of the equation.

Step 3: To graph the equation, start by plotting the y-intercept on the y-axis. Since the y-intercept is 0, put a point at the origin (0,0).

Step 4: From the y-intercept, use the slope to find another point on the line. The slope is 1/3, which means that for every 1 unit you move up on the y-axis, you move 3 units to the right on the x-axis. So, from the point at the origin, move 1 unit up and 3 units to the right and put another point.

Step 5: Draw a line through the two points. This line represents the graph of the equation y = x/3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Equations

A linear equation is an algebraic expression that represents a straight line when graphed on a coordinate plane. It typically takes the form y = mx + b, where m is the slope and b is the y-intercept. Understanding linear equations is essential for graphing, as it allows one to identify the relationship between the variables involved.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept. This form is particularly useful for quickly identifying how steep the line is and where it crosses the y-axis, facilitating easier graphing of the equation.

Graphing Techniques

Graphing techniques involve methods for accurately plotting equations on a coordinate plane. For linear equations, one can find key points such as the y-intercept and use the slope to determine additional points. Mastery of these techniques is crucial for visualizing mathematical relationships and interpreting the behavior of functions.