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

2. Graphs of Equations

Two-Variable Equations

College Algebra

2. Graphs of Equations

Two-Variable Equations

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1:48 minutes

Problem 1

Textbook Question

Graph each equation in Exercises 1–4. Let x= -3, -2. -1, 0, 1, 2 and 3. y = 2x-2

Verified step by step guidance

Step 1: Understand the equation y = 2x - 2. This is a linear equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

Step 2: Create a table of values for x and y. Use the given x-values (-3, -2, -1, 0, 1, 2, 3). For each x-value, substitute it into the equation y = 2x - 2 to calculate the corresponding y-value.

Step 3: For example, when x = -3, substitute into the equation: y = 2(-3) - 2. Similarly, calculate y for all other x-values (-2, -1, 0, 1, 2, 3).

Step 4: Once you have the table of x and y values, plot these points on a coordinate plane. Each pair (x, y) represents a point on the graph.

Step 5: Draw a straight line through the plotted points, as the equation represents a linear function. Ensure the line extends in both directions and label the graph appropriately.

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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 students to identify how changes in x affect y.

Slope and Y-Intercept

The slope of a linear equation indicates the steepness and direction of the line, calculated as the change in y over the change in x (rise/run). The y-intercept is the point where the line crosses the y-axis, representing the value of y when x is zero. In the equation y = 2x - 2, the slope is 2 and the y-intercept is -2, which are crucial for accurately plotting the graph.

Graphing Points

Graphing points involves plotting specific (x, y) coordinates on a Cartesian plane. For the equation y = 2x - 2, students will substitute the given x-values (-3, -2, -1, 0, 1, 2, 3) to find corresponding y-values. This process helps visualize the relationship between x and y, forming the linear graph of the equation.