Textbook Question
Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² - 6y -7=0
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 3, Problem 60
In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.
Verified step by step guidance1
Recall the vertical line test: A graph represents y as a function of x if and only if no vertical line intersects the graph at more than one point.
Observe the given graph, which is a straight line sloping downward crossing both the x-axis and y-axis.
Imagine drawing vertical lines (lines parallel to the y-axis) at various x-values along the graph.
Check if any vertical line intersects the graph at more than one point. Since this is a straight line, each vertical line will intersect it at exactly one point.
Conclude that since every vertical line intersects the graph at only one point, the graph passes the vertical line test, so y is a function of x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertical Line Test
The vertical line test is a method used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function. This test ensures that each input (x-value) has only one output (y-value).
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The Slope of a Line
Definition of a Function
A function is a relation where each input value (x) corresponds to exactly one output value (y). This means no x-value can be paired with multiple y-values. Understanding this definition is crucial to identifying functions from graphs.
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5:57Graphs of Common Functions
Graph of a Linear Function
A linear function is represented by a straight line on the coordinate plane. Since a line passes the vertical line test (no vertical line intersects it more than once), it always represents a function. The graph shown is a downward sloping line, indicating a linear function with a negative slope.
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5:26Graphs of Logarithmic Functions
Related Practice
Textbook Question
In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x^2. Then use transformations of this graph to graph the given function. g(x) = x^2 + 2
Textbook Question
Use the vertical line test to identify graphs in which y is a function of x.
Textbook Question
Find a. (fog) (x) b. (go f) (x) f(x) = 4-x, g(x) = 2x² +x+5
Textbook Question
Find a. (fog) (2) b. (go f) (2) f(x) = 4-x, g(x) = 2x² +x+5
Textbook Question
In Exercises 59-66, a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. 4x+y-6=0
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