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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 53
Chapter 3, Problem 53

For each line, (a) find the slope and (b) sketch the graph. 2y = -3x

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Rewrite the given equation in slope-intercept form, which is \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. Start by isolating \(y\) in the equation \(2y = -3x\) by dividing both sides by 2, resulting in \(y = \frac{-3}{2}x\).
Identify the slope \(m\) from the equation \(y = \frac{-3}{2}x\). Here, the slope is \(m = \frac{-3}{2}\), which means for every 2 units you move horizontally to the right, the line moves 3 units down vertically.
Note that the y-intercept \(b\) is 0 in this equation, meaning the line passes through the origin \((0,0)\).
To sketch the graph, start by plotting the y-intercept point at \((0,0)\) on the coordinate plane.
From the y-intercept, use the slope \(\frac{-3}{2}\) to find another point: move 2 units to the right (positive x-direction) and 3 units down (negative y-direction), then plot this second point. Draw a straight line through these two points to complete the graph.

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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 measures the steepness of a line and is calculated as the ratio of the change in y to the change in x (rise over run). For a line in the form y = mx + b, the slope is m. Understanding slope helps determine how the line inclines or declines on a graph.
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The Slope of a Line

Rearranging Linear Equations to Slope-Intercept Form

To find the slope easily, rewrite the equation in slope-intercept form (y = mx + b). This involves isolating y on one side. For example, from 2y = -3x, dividing both sides by 2 gives y = (-3/2)x, revealing the slope as -3/2.
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Slope-Intercept Form

Graphing a Line Using Slope and Intercept

Graphing a line involves plotting the y-intercept and using the slope to find other points. Starting at the intercept, move vertically and horizontally according to the slope's rise and run. This method provides a visual representation of the linear equation.
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Graphing Lines in Slope-Intercept Form
Related Practice
Textbook Question

Find the value of the function for the given value of x.

f(x)={3if 0<x≤410−2[[5−x]]if x>4, for , for x=6.2f(x)=\(\begin{cases}\)3 & \(\text{if }\)0<x\(\leq\)4\\ 10-2[\(\left\]\lbrack\)5-x]\(\right\[\rbrack\) & \(\text{if }\)x>4,\(\text{ for }\]\end{cases}\),\(\text{ for }\)x=6.2

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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)=−x2ƒ(x)=-x^2

Textbook Question

Find the slope of each line, provided that it has a slope. 11x + 2y = 3

Textbook Question

Let ƒ(x)=-3x+4 and g(x)=-x2+4x+1. Find each of the following. Simplify if necessary. g(-2)

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Textbook Question

Write an equation (a) in standard form and (b) in slope-intercept form for each line described. through (1, 6), perpendicular to 3x+5y=1

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Textbook Question

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