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Chapter 3: Problem 84

If the graph of a linear equation rises from left to right, then the average rate of change is _____(positive/negative). If the graph of a linear equation falls from left to right, then the average rate of change is _____(positive/negative).

Short Answer

Expert verified
Positive; Negative

Step by step solution

01

Title - Understand the Problem

To determine whether the average rate of change is positive or negative, analyze the direction in which the graph is moving from left to right.
02

Title - Determine the Direction

If a graph rises from left to right, it is moving upward in the direction of positive y-values as x-values increase.
03

Title - Positive Slope

When the graph rises from left to right, the slope is positive. Hence, the average rate of change is positive.
04

Title - Falls from Left to Right

If a graph falls from left to right, it is moving downward in the direction of negative y-values as x-values increase.
05

Title - Negative Slope

When the graph falls from left to right, the slope is negative. Hence, the average rate of change is negative.

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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 mathematical statements that show a relationship between two variables with consistent rates of change. Usually, these equations appear in the form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) tells us how steep the line is and in which direction it goes. A line can either rise, fall, or be flat. Understanding linear equations is crucial because they form the foundation for graphing and solving real-life problems. Start with identifying the slope to determine the direction of the line.
Positive Slope
Imagine you are looking at a graph where a line rises as you move from the left side to the right side. This situation represents a positive slope. When the slope is positive, the value of \( m \) in the linear equation \( y = mx + b \) will be greater than zero. A positive slope means that as \( x \) increases, \( y \) also increases.
  • If the slope is \( m = 2 \), it means for every unit increase in \( x \), \( y \) increases by 2 units.
  • A real-life example might be a car speeding up, where the distance traveled increases as time goes on.
Recognize that a positive slope graphically indicates an upward trend, useful for identifying increasing trends or growth in various applications.
Negative Slope
Now let's consider a graph where the line falls as you move from the left side to the right side. This situation represents a negative slope. When the slope is negative, the value of \( m \) in the linear equation \( y = mx + b \) will be less than zero. A negative slope means that as \( x \) increases, \( y \) decreases.
  • If the slope is \( m = -3 \), it means for every unit increase in \( x \), \( y \) decreases by 3 units.
  • A real-life example might be a car slowing down or coming to a stop, where the speed decreases over time.
Recognizing a negative slope visually indicates a downward trend, which is crucial for understanding declining trends or losses in various scenarios.
Graph Analysis
Graph analysis involves interpreting the information shown on a graph to understand the relationship between the variables. For linear equations, the primary feature analyzed is the slope. To analyze a graph:
  • First, identify whether the line goes upward or downward as you move from left to right.

  • If the line rises, identify that it has a positive slope.
  • If the line falls, determine that it has a negative slope.
Graph analysis helps you understand real-world situations modeled by linear equations. For example, in economics, you might graph supply and demand curves to see how they interact. Natural sciences often use graph analysis to show growth rates of populations or the speed of chemical reactions. Clear interpretation and understanding of graph features like slope facilitate better decision-making and problem-solving.

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Most popular questions from this chapter

For each function, find \((a) f(2)\) and \((b) f(-1) .\) See Examples 4 and \(5 .\) $$ f=\\{(2,5),(3,9),(-1,11),(5,3)\\} $$

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ \left(-\frac{4}{9},-6\right) \text { and }\left(\frac{12}{7},-6\right) $$

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ -2 x+5 y=9 $$

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ (7,6) \text { and }(7,-8) $$

Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Find the slope of segment \(A C\).
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