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Chapter 3: Problem 92

Find the slope of the line through the given points. The graph of \(y=-\frac{1}{3} x+2\) has a slope of \(-\frac{1}{3}\). The graph of \(y=-2 x+2\) has a slope of -2 . The graph of \(y=-4 x+2\) has a slope of -4 . Graph all three equations on a single coordinate system. As the absolute value of the slope becomes larger, how does the steepness of the line change?

Short Answer

Expert verified
As the absolute value of the slope increases, the steepness of the line increases.

Step by step solution

01

Identify the Slope of Each Line

Each line is represented by the equation \[ y = mx + b \] where \( m \) is the slope. The given equations are: 1. \( y = -\frac{1}{3}x + 2 \) with a slope of \( -\frac{1}{3} \).2. \( y = -2x + 2 \) with a slope of \( -2 \). 3. \( y = -4x + 2 \) with a slope of \( -4 \). Thus, the slopes are \( -\frac{1}{3}, -2, \) and \( -4 \) respectively.
02

Understand the Relationship Between Slope and Steepness

The slope \( m \) indicates how steep the line is. A larger absolute value of \( m \) means a steeper line. Comparing the slopes,- \( -\frac{1}{3} \) is the least steep,- \( -2 \) is steeper than \( -\frac{1}{3} \),- \( -4 \) is the steepest line among the three.
03

Graph Each Equation

Plot each line on the same set of axes. Start by plotting the y-intercept (\( b = 2 \) for all lines) at the point (0,2).- For \( y = -\frac{1}{3}x + 2 \), move down 1 unit and right 3 units from the y-intercept for the next point.- For \( y = -2x + 2 \), move down 2 units and right 1 unit from the y-intercept for the next point.- For \( y = -4x + 2 \), move down 4 units and right 1 unit from the y-intercept for the next point. Draw each line through these points.

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Key Concepts

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

Linear Equations
Linear equations are fundamental in understanding the relationship between variables. They are usually written in the form \[ y = mx + b \] where:
  • \( y \) is the dependent variable,
  • \( x \) is the independent variable,
  • \( m \) is the slope of the line,
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
In linear equations, the value of \( m \) determines how the line slants as you move along the x-axis. If \( m \) is positive, the line goes up; if it's negative, the line goes down. The y-intercept, \( b \), helps you identify the starting point on the y-axis. Whenever you encounter a linear equation, break it down by first finding \( m \) and \( b \). This makes it easier to graph the line and understand its behavior.
Graphing a Line
Graphing a line from a linear equation involves understanding its slope and y-intercept. To graph the equation \[ y = mx + b \], follow these steps:
  • Start by plotting the y-intercept \( b \) on the y-axis. This is the point \((0, b)\).
  • Use the slope \( m \) to determine the direction of the line. The slope \( m \) is often written as a fraction: \( \frac{rise}{run} \).
  • From the y-intercept, move vertically by the "rise" value and horizontally by the "run" value.
  • Mark this new point and draw a line through the y-intercept and this point.
For example, in the equation \( y = -2x + 2 \), start at the y-intercept (0,2). From there, the slope \(-2\) tells you to move down 2 units and right 1 unit to find your next point. Connect these points with a straight line and continue it across the grid to complete your graph.
Linear Functions
Linear functions are a type of function that produce straight lines when graphed. They are crucial in math because they form the basis for understanding more complex functions. Here's why linear functions are important:
  • They provide a simple model to show constant rates of change, which means for every consistent change in \( x \), \( y \) changes by a fixed amount, determined by the slope.
  • They can be used to predict values outside the initial data set within a reasonable range.
  • Understanding linear functions helps in financial projections, physics problems, and any scenario involving uniform rates.
For example, the equation \( y = -\frac{1}{3}x + 2 \) describes a linear function where for every 3 units increase in \( x \), \( y \) decreases by 1 unit. It represents a constant, predictable relationship between \( x \) and \( y \). By mastering these functions, you gain a powerful tool for mathematical modeling.

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Most popular questions from this chapter

Find an equation of each line described. Write each equation in slope- intercept form when possible. Through \((1,2),\) parallel to \(y=5\)

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In \(2003,\) there were 302 million magazine subscriptions in the United States. By 2007 , this number was 322 million. (Source: Audit Bureau of Circulation, Magazine Publishers Association) a. Write two ordered pairs of the form (years after \(2003,\) millions of magazine subscriptions) for this situation. b. Assume the relationship between years after 2003 and millions of magazine subscriptions is linear over this period. Use the ordered pairs from part (a) to write an equation for the line relating year after 2003 to millions of magazine subscriptions. C. Use this linear equation in part (b) to estimate the millions of magazine subscriptions in 2005 .

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In \(2006,\) the U.S. population per square mile of land area was approximately \(83.6 .\) In 2000 , the population per square mile was 79.6 a. Assume the relationship between years past 2000 and population per square mile is linear over this period. Write an equation describing the relationship between year and population per square mile. Use ordered pairs of the form (years past 2000 , population per square mile). b. Use this equation to predict the population per square mile in 2010 .

Solve. See Example 4. The table shows the amount of money (in billions of dollars) that Americans spent on their pets for the years shown. (Source: American Pet Products Manufacturers Association) $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Pet-Related Expenditures } \\ \text { (in billions of dollars) } \end{array} \\ \hline 2005 & 36.3 \\ \hline 2006 & 38.5 \\ \hline 2007 & 41.2 \\ \hline 2008 & 43.4 \\ \hline \end{array} $$ a. Write this paired data as a set of ordered pairs of the form (year, pet- related expenditures). b. In your own words, write the meaning of the ordered pair (2007,41.2) c. Create a scatter diagram of the paired data. \(\mathrm{Be}\) sure to label the axes appropriately. d. What trend in the paired data does the scatter diagram show?

Find the value of \(x^{2}-3 x+1\) for each given value of \(x\). $$ -1 $$
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