Textbook Question
Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. (4, -1) and (3, −1)
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2. Graphs of Equations
Lines
3:36 minutesProblem 17
Textbook Question
Find the average rate of change of the function from x1 to x2. f(x) = √x from x1 = 4 to x2 = 9
Verified step by step guidance1
Identify the function given: \(f(x) = \sqrt{x}\), and the interval endpoints \(x_1 = 4\) and \(x_2 = 9\).
Recall the formula for the average rate of change of a function \(f(x)\) over the interval \([x_1, x_2]\):
\(\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\).
Calculate the function values at the endpoints:
\(f(x_1) = f(4) = \sqrt{4}\) and \(f(x_2) = f(9) = \sqrt{9}\).
Substitute these values into the average rate of change formula:
\(\frac{\sqrt{9} - \sqrt{4}}{9 - 4}\).
Simplify the numerator and denominator separately, then divide to find the average rate of change over the interval.
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Video duration:
3mKey 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 between two points measures how much the function's output changes per unit change in input. It is calculated as the difference in function values divided by the difference in input values, similar to the slope of a secant line connecting the two points.
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Change of Base Property
Square Root Function
The square root function, denoted as f(x) = √x, outputs the non-negative number whose square is x. It is defined for x ≥ 0 and is a common example of a radical function, which often requires careful evaluation when calculating function values at specific points.
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Evaluating Functions at Given Points
To find the average rate of change, you must correctly evaluate the function at the specified input values. This involves substituting the input values into the function and simplifying, ensuring accurate calculation of the function's outputs at those points.
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