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

Graphs and Coordinates

College Algebra

2. Graphs of Equations

Graphs and Coordinates

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1:22 minutes

Problem 86

Textbook Question

In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).] y = √(x - 4) + √(x + 4) - 4

Verified step by step guidance

Step 1: Set the equation equal to zero to find the x-intercepts. This means solving the equation \( \sqrt{x - 4} + \sqrt{x + 4} - 4 = 0 \).

Step 2: Isolate one of the square root terms. For example, move \( \sqrt{x + 4} \) to the other side: \( \sqrt{x - 4} = 4 - \sqrt{x + 4} \).

Step 3: Square both sides of the equation to eliminate the square root on the left side: \( x - 4 = (4 - \sqrt{x + 4})^2 \).

Step 4: Expand the right side of the equation: \( x - 4 = 16 - 8\sqrt{x + 4} + (x + 4) \).

Step 5: Simplify the equation and solve for \( x \) by isolating the square root term again and squaring both sides if necessary.

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Key Concepts

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

X-Intercepts

X-intercepts are the points where a graph intersects the x-axis, which occurs when the value of y is zero. To find the x-intercepts of an equation, you set y equal to zero and solve for x. This is crucial for understanding the behavior of the graph and identifying key points that help in graphing the function.

Square Root Functions

Square root functions, such as y = √(x - a), are defined for values of x that make the expression under the square root non-negative. This means that the domain of the function is limited to x values greater than or equal to a. Understanding the domain is essential for determining where the function is valid and for finding x-intercepts.

Graph Matching

Graph matching involves comparing the characteristics of a function's graph with given options to identify the correct representation. This requires understanding the shape and key features of the graph, such as intercepts, asymptotes, and overall behavior. By analyzing the x-intercepts and other properties, one can effectively match the equation to its corresponding graph.