Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 49

Chapter 3, Problem 49

Find the slope of each line, provided that it has a slope. through (2, -2) and (3, -4)

Verified step by step guidance

Recall that the slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Identify the coordinates of the two points given: Point 1: $(2, -2)$ so $x_1 = 2$ and $y_1 = -2$ Point 2: $(3, -4)$ so $x_2 = 3$ and $y_2 = -4$

Substitute the values into the slope formula: \[m = \frac{-4 - (-2)}{3 - 2}\]

Simplify the numerator and denominator separately: Numerator: $-4 - (-2) = -4 + 2$ Denominator: $3 - 2 = 1$

Write the simplified fraction for the slope: \[m = \frac{-4 + 2}{1}\] This fraction represents the slope of the line through the two points.

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Key Concepts

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

Slope of a Line

The slope of a line measures its steepness and direction, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points. It is given by the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are coordinates of the points.

Coordinate Points

Coordinate points represent specific locations on the Cartesian plane, expressed as (x, y). Understanding how to use these points is essential for calculating slope, as the differences in their x and y values determine the line's steepness.

Undefined Slope

A line has an undefined slope when the change in x (run) is zero, meaning the line is vertical. In such cases, the slope formula results in division by zero, indicating no defined slope exists for vertical lines.

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Related Practice

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Find the value of the function for the given value of x. ƒ(x)=[[3-(x/2)]], for x=1

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Work each of the following. Find the equation of a circle with center at (-4, 3), passing through the point (5, 8).Write it in center-radius form.

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The graph of a linear function f is shown. (a) Identify the slope, y-intercept, and x-intercept. (b) Write an equation that defines f.

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For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=1/x

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For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. 2x+3y=5

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Textbook Question

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4.

$ƒ (x) = - 4 x + 2$