Textbook Question
For each function, find (a) ƒ(2) and (b) ƒ(-1).See Example 7. ƒ = {(2,5),(3,9),(-1,11),(5,3)}
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Ch. 2 - Graphs and Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 3, Problem 68
Find and interpret the average rate of change illustrated in each graph.
Verified step by step guidance1
Identify two points on the graph to calculate the average rate of change. For example, use the points (0, 135) and (8, 135).
Recall the formula for the average rate of change between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}\]
Substitute the coordinates of the chosen points into the formula:
\[\frac{135 - 135}{8 - 0}\]
Simplify the numerator and denominator separately to find the average rate of change:
\[\frac{0}{8}\]
Interpret the result: since the average rate of change is zero, this means the function's value does not change as \(x\) changes, indicating a constant function represented by a horizontal line.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Average Rate of Change
The average rate of change of a function over an interval measures how much the function's output changes per unit change in input. It is calculated as the change in the y-values divided by the change in the x-values between two points on the graph.
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Change of Base Property
Interpreting Horizontal Lines
A horizontal line on a graph indicates that the function's output remains constant regardless of changes in the input. This means the average rate of change is zero, as there is no increase or decrease in the y-values over the interval.
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06:49The Slope of a Line
Slope of a Line
The slope of a line represents the rate at which y changes with respect to x. For a horizontal line, the slope is zero, indicating no change in y-values. Understanding slope helps interpret the average rate of change visually and numerically.
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Related Practice
Textbook Question
Graph each function.
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Textbook Question
For each line described, write an equation in (a) slope-intercept form, if possible, and (b) standard form. through (3, -5), parallel to y = 4
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Textbook Question
Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=x2+2
Textbook Question
Graph each function.
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Textbook Question
Solve each problem. A graph of y=ƒ(x) is shown in the standard viewing window. Which is the only value of x that could possibly be the solution of the equation ƒ(x) =0? A. -15 B. 0 C. 5 D. 15