Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 19

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 guidance

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.

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.

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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