Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Question:** Which of the following describes the graph of \( h(x) = -2^{(x-3)} + 4 \)? **Graphs:** 1. **First Graph:** - The graph consists of a curve that starts from the top-left corner, decreasing rapidly as it moves to the right. - It shows a near-vertical descent and gradually flattens out as it approaches \( y = -16 \) at \( x = 9 \). - The curve closely follows the y-axis below before turning towards the right. 2. **Second Graph:** - The graph begins from the bottom-left corner and moves upwards to the right. - Initially, the curve rises steeply and then starts to flatten out as it progresses rightwards. - The curve approaches but does not appear to cross \( y = 4 \) as it continues to move to the right. This suggests a horizontal asymptote at \( y = 4 \). - The graph levels off near \( y = 4 \) as \( x \) moves beyond \( 9 \), exemplifying the behavior of the exponential function \( -2^{(x-3)} + 4 \). **Explanation:** The graphs provided are to be analyzed to determine which one correctly represents the function \( h(x) = -2^{(x-3)} + 4 \). The second graph suits the characteristics of this exponential function since it reflects the behavior expected — rapidly increasing and asymptotically approaching \( y = 4 \) from below.
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Transcribed Image Text:**Question:** Which of the following describes the graph of \( h(x) = -2^{(x-3)} + 4 \)? **Graphs:** 1. **First Graph:** - The graph consists of a curve that starts from the top-left corner, decreasing rapidly as it moves to the right. - It shows a near-vertical descent and gradually flattens out as it approaches \( y = -16 \) at \( x = 9 \). - The curve closely follows the y-axis below before turning towards the right. 2. **Second Graph:** - The graph begins from the bottom-left corner and moves upwards to the right. - Initially, the curve rises steeply and then starts to flatten out as it progresses rightwards. - The curve approaches but does not appear to cross \( y = 4 \) as it continues to move to the right. This suggests a horizontal asymptote at \( y = 4 \). - The graph levels off near \( y = 4 \) as \( x \) moves beyond \( 9 \), exemplifying the behavior of the exponential function \( -2^{(x-3)} + 4 \). **Explanation:** The graphs provided are to be analyzed to determine which one correctly represents the function \( h(x) = -2^{(x-3)} + 4 \). The second graph suits the characteristics of this exponential function since it reflects the behavior expected — rapidly increasing and asymptotically approaching \( y = 4 \) from below.
The image contains two graphs, each displaying a function \( h(x) \). #### First Graph (Top): - **Axes**: - The horizontal axis (x-axis) ranges from -9 to 9. - The vertical axis (y-axis) ranges from -16 to 2. - **Function**: \( h(x) \) - **Behavior**: - As \( x \) increases from left to right, the value of \( h(x) \) decreases sharply, continuing into the negative \( y \) values. - The graph appears to taper off more sharply as \( x \) becomes more positive, suggesting a steep decline. #### Second Graph (Bottom): - **Axes**: - The horizontal axis (x-axis) ranges from -9 to 9. - The vertical axis (y-axis) ranges from -16 to 2. - **Function**: \( h(x) \) - **Behavior**: - As \( x \) increases from left to right, the value of \( h(x) \) increases sharply from large negative \( y \) values to higher \( y \) values. - The graph shows a steep incline as \( x \) becomes more positive. ### Comparison: - Both graphs feature the same function, \( h(x) \), observed over a consistent range for \( x \) and \( y \) values. ### Purpose: These graphs can be used to illustrate different behaviors of a mathematical function, such as exponential decay in the first graph and exponential growth in the second graph. They serve as excellent visuals for understanding how function transformations reflect over the y-axis or around different points.
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Transcribed Image Text:The image contains two graphs, each displaying a function \( h(x) \). #### First Graph (Top): - **Axes**: - The horizontal axis (x-axis) ranges from -9 to 9. - The vertical axis (y-axis) ranges from -16 to 2. - **Function**: \( h(x) \) - **Behavior**: - As \( x \) increases from left to right, the value of \( h(x) \) decreases sharply, continuing into the negative \( y \) values. - The graph appears to taper off more sharply as \( x \) becomes more positive, suggesting a steep decline. #### Second Graph (Bottom): - **Axes**: - The horizontal axis (x-axis) ranges from -9 to 9. - The vertical axis (y-axis) ranges from -16 to 2. - **Function**: \( h(x) \) - **Behavior**: - As \( x \) increases from left to right, the value of \( h(x) \) increases sharply from large negative \( y \) values to higher \( y \) values. - The graph shows a steep incline as \( x \) becomes more positive. ### Comparison: - Both graphs feature the same function, \( h(x) \), observed over a consistent range for \( x \) and \( y \) values. ### Purpose: These graphs can be used to illustrate different behaviors of a mathematical function, such as exponential decay in the first graph and exponential growth in the second graph. They serve as excellent visuals for understanding how function transformations reflect over the y-axis or around different points.
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