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Chapter 4: Problem 3

Describe the slant of a line with a negative slope.

Short Answer

Expert verified
A line with a negative slope slants downwards from left to right, meaning the y-values decrease as the x-values increase. In the equation \(y = mx + b\), the slope (m) will be a negative value. The greater the absolute value of the negative slope, the steeper the downward slant.

Step by step solution

01

Understanding slope

Slope is a measure of the steepness of a line, and it can be found by calculating the difference in the y-values (vertical distance) divided by the difference in the x-values (horizontal distance) of two points on the line. This is represented by the formula: slope (m) = \(\frac{y2 - y1}{x2 - x1}\). Importantly, the slope can be positive, negative, or zero, and this will affect the slant of the line.
02

Description of a negative slope

A line with a negative slope is one where the line slants downwards from left to right. As the x-values increase, the y-values decrease. In other words, when moving from left to right along the line, the line goes down in the graph. The greater the absolute value of the negative slope, the steeper the downward slant.
03

Equation of a line with a negative slope

A line with a negative slope can be represented by the equation: \(y = mx + b\), where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). Since the slope is negative, m will be a negative value in the equation. For example, if the slope is -2 and the y-intercept is 3, the equation of the line will be: \(y = -2x + 3\).
04

Visual representation of a line with a negative slope

To draw a line with a negative slope, start at the y-intercept (b) and then move down and to the right according to the slope (m). For instance, using the example above where the slope is -2 and the y-intercept is 3, start at the point (0, 3) on the graph. Since the slope is -2 (which can be written as \(\frac{-2}{1}\)), move down 2 units and to the right 1 unit for the next point. Keep following this pattern to draw the line. Notice that the line slants downwards from left to right.

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Most popular questions from this chapter

Graph each function using the slope and \(y\) -intercept. $$f(x)=-x+5$$

Graph each function using the slope and \(y\) -intercept. $$h(x)=3 x+1$$

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{3}{x-5}$$

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope-intercept form or in standard form, as indicated. $$y=\frac{1}{4} x-7 ;(-2,7) ; \text { standard form }$$

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{1}{-6+4 x}$$
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