Textbook Question
In Exercises 11–26, determine whether each equation defines y as a function of x. 4x = y²
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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 19
Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ passes through (−1, 5) and is perpendicular to the line whose equation is x = 6.
Verified step by step guidance1
Identify the slope of the given line. The line given is \( x = 6 \), which is a vertical line. Vertical lines have an undefined slope.
Determine the slope of the line perpendicular to the given line. Since the given line is vertical, the perpendicular line must be horizontal, which means its slope is \( 0 \).
Use the point-slope form of a line equation with the slope \( m = 0 \) and the point \( (-1, 5) \). The point-slope form is \( y - y_1 = m(x - x_1) \).
Substitute the values into the point-slope form: \( y - 5 = 0 \times (x + 1) \). Simplify the right side to get \( y - 5 = 0 \).
Solve for \( y \) to write the equation in slope-intercept form \( y = mx + b \). This gives \( y = 5 \), which is the equation of the line perpendicular to \( x = 6 \) passing through \( (-1, 5) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Slope-Intercept Form
The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b is the y-intercept. This form makes it easy to identify the slope and where the line crosses the y-axis, which is essential for graphing and writing linear equations.
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03:56
Slope-Intercept Form
Perpendicular Lines
Two lines are perpendicular if their slopes are negative reciprocals of each other. For vertical and horizontal lines, a vertical line has an undefined slope, and a line perpendicular to it is horizontal with a slope of zero. Understanding this helps determine the slope of the required line.
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07:52Parallel & Perpendicular Lines
Using a Point to Find the Equation
Given a point on the line and the slope, you can substitute these values into the slope-intercept form to solve for the y-intercept b. This step is crucial to write the specific equation of the line that meets the given conditions.
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04:27Finding Equations of Lines Given Two Points
Related Practice
Textbook Question
Use the graph to determine (a) the function's domain, (b) the function's range, (c) the x-intercepts, if any, (d) the y-intercept, if there is one, (e) intervals on which the function is increasing, decreasing or constant, (f) the missing function values, indicated by question marks, below each graph.
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Textbook Question
use the graph of y = f(x) to graph each function g.
g(x) = f(x-1)
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Textbook Question
Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (-1/4, -1/7) and (3/4, 6/7)
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Textbook Question
Find the midpoint of each line segment with the given endpoints. (6, 8) and (2, 4)
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Textbook Question
The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ-1 (x)) = = x and ƒ-1 (f(x)) = x. f(x) = (x+2)³
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