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Chapter 1: Problem 142

The slopes of two lines are -4 and \(\frac{5}{2}\). Which is steeper? Explain.

Short Answer

Expert verified
The line with a slope of -4 is steeper.

Step by step solution

01

Understand the concept of slope

The slope of a line in the plane containing the x and y axes is generally represented by the letter 'm', and it is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. More simply, slope is the measure of how steep a line is.
02

Identify the given slopes

The slopes of the two lines are given as -4 and \(\frac{5}{2}\). These values are what need to be compared.
03

Compare the absolute values of the slopes

The absolute value of -4 is 4 and the absolute value of \(\frac{5}{2}\) is \(\frac{5}{2}\). Comparing these values, 4 is greater than \(\frac{5}{2}\). So, the line with the greater absolute value of the slope is steeper.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Steepness of a line
When we talk about the steepness of a line, we are referring to how sharply it rises or falls as you move along the x-axis. Think of slope like a ramp or a hill. The steeper the ramp, the harder it is to climb. Mathematically, the steepness is described by a number called the slope.
  • If the slope is positive, it means the line is rising from left to right.
  • If the slope is negative, the line is falling from left to right.
  • A larger absolute value of the slope indicates a steeper line, regardless of whether it's positive or negative.
To find the slope between any two points a \ "point on the line, you can use the formula: \ m = \frac{y_2 - y_1}{x_2 - x_1} \ where \(y_2 - y_1\) is the vertical change and \(x_2 - x_1\) is the horizontal change between two points \((x_1, y_1)\) and \((x_2, y_2)\). This formula gives you a practical tool for calculating steepness.
Absolute value
The absolute value is a key concept to understand when comparing slopes. Absolute value refers to how far a number is from zero on the number line, regardless of direction.
  • For any real number \(a\), the absolute value is represented as \(|a|\).
  • For positive numbers, the absolute value is the number itself, like \(|5| = 5\).
  • For negative numbers, it is their positive counterpart, such as \(|-4| = 4\).
In the context of slopes, this is important because the absolute value ignores the direction of the slope's steepness (upward or downward). It helps in figuring out the slope's magnitude鈥攈ow steep the line is鈥攚ithout concern for its direction.
Comparing slopes
When comparing slopes to determine which line is steeper, you focus on the magnitude of their steepness. You can achieve this effectively by using the absolute values of the slopes.
  • Consider slopes -4 and \(\frac{5}{2}\).
  • First, find the absolute values: \|-4| = 4 and \|\frac{5}{2}| = \frac{5}{2}\.
  • Compare these absolute values: 4 and \(\frac{5}{2}\).
Since 4 is greater than \(\frac{5}{2}\), the line with a slope of -4 is steeper than the line with a slope of \(\frac{5}{2}\). This method of comparing absolute values allows you to determine the relative steepness of lines in an efficient manner.

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Most popular questions from this chapter

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\sqrt[3]{x-1} $$

Write a sentence using the variation terminology of this section to describe the formula. Area of a triangle: \(A=\frac{1}{2} b h\)

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