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Magazine Chapter 6: Problem 24
Graph the function and its reflection about the x-axis on the same axes. $$f(x)=3(0.75)^{x}-1$$
Short Answer
Expert verified
Graph \( f(x) = 3(0.75)^x - 1 \) and its reflection \(-f(x) = -3(0.75)^x + 1\).
Step by step solution
01
Understand the Given Function
The function provided is \( f(x) = 3(0.75)^{x} - 1 \). This is an exponential function where the base is less than 1, indicating a decreasing exponential behavior. The constant term \(-1\) shifts the whole graph downward by 1 unit.
02
Find Key Points of the Original Function
To graph the function, find key values using selected \( x \) values. For example:- \( f(0) = 3(0.75)^0 - 1 = 2 \)- \( f(1) = 3(0.75)^1 - 1 = 3(0.75) - 1 = 1.25 \)- \( f(2) = 3(0.75)^2 - 1 = 3(0.5625) - 1 = 0.6875 \)Calculate more points if needed for accuracy.
03
Graph the Original Function
Plot the points from Step 2 on the coordinate plane and connect them smoothly, showing the decreasing nature of the function due to the exponential decay of \((0.75)^x\). The graph should pass through points (0, 2), (1, 1.25), (2, 0.6875), approaching the horizontal line \( y = -1 \) as \( x \) increases.
04
Determine the Reflection Function
The reflection of a function about the x-axis involves multiplying the function by \(-1\). Thus, the reflection of \( f(x) = 3(0.75)^x - 1 \) is \( -f(x) = -(3(0.75)^x - 1) = -3(0.75)^x + 1 \).
05
Find Key Points of the Reflection Function
Evaluate points for the reflection using the same \( x \) values used previously:- \(-f(0) = -3(0.75)^0 + 1 = -2 \)- \(-f(1) = -3(0.75)^1 + 1 = -1.25 \)- \(-f(2) = -3(0.75)^2 + 1 = -0.6875 \)These points show the behavior of the reflection.
06
Graph the Reflection Function
Plot the points from Step 5 on the same coordinate plane used in Step 3. Connect these points, forming a mirror image of the original graph across the x-axis. The graph should pass through (0, -2), (1, -1.25), (2, -0.6875), moving away from the horizontal line \( y = 1 \) as \( x \) increases.
07
Review and Compare
Look at both graphs on the same coordinate plane. The original function decreases and approaches \( y = -1 \), while the reflection increases from \( y = -3 \) to \( y = 1 \) and diverges upward.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential decay
Exponential decay refers to the process where a quantity decreases over time at a rate proportional to its current value. In the function \( f(x) = 3(0.75)^x - 1 \), the base \( 0.75 \) is less than 1. This indicates that the function experiences exponential decay as \( x \) increases. For each increase in \( x \), the factor \((0.75)^x\) becomes smaller, reducing the overall value of \( f(x) \).
Key Characteristics of Exponential Decay:
Key Characteristics of Exponential Decay:
- The graph starts at a higher value when \( x = 0 \) and decreases as \( x \) increases.
- The decay is rapid at first but slows over time, reflecting a common pattern seen in natural decaying processes such as radioactive decay or cooling of hot objects.
- In the provided function, the term \(-1\) shifts the entire graph downward by 1 unit, affecting the overall asymptotic behavior, essentially setting the horizontal asymptote at \( y = -1 \). This means the function will never actually reach \( y = -1 \), but instead it will get infinitely close as \( x \) increases.
Reflection over x-axis
Reflecting a graph over the x-axis involves creating a mirror image of the original function across this axis, effectively flipping it upside down. If you imagine the x-axis as a mirror, every point on the graph is flipped. For the function \( f(x) = 3(0.75)^x - 1 \), its reflection is calculated by taking the negative of the function: \(-f(x) = -3(0.75)^x + 1 \).
- This transformation changes every value of \( f(x) \) to its opposite: if a value is positive, the reflection will be negative, and vice versa.
- For example, a point located at \( (x, 2) \) on the original graph will be at \( (x, -2) \) on the reflected graph.
- The process of reflection over the x-axis does not affect the x-coordinates of the points, only the y-values change.
- The reflection graph provides a visual representation of how the original function behaves when all outputs are inverted.
- In our exercise, while the original function grapples with exponential decay moving toward \( y = -1 \), its reflection graph will show exponential growth but negatively, moving away from \( y = 1 \).
Coordinate plane
The coordinate plane is a fundamental tool in mathematics used for graphing functions and understanding their behaviors. It consists of two perpendicular lines: the horizontal x-axis and the vertical y-axis. Each point on the plane is defined by a pair of coordinates \((x, y)\).
Importance of the Coordinate Plane:
Importance of the Coordinate Plane:
- It allows us to visually represent mathematical functions, making it easier to understand their properties and graph behaviors.
- By plotting points on this plane, we can connect them to form the overall shape of a graph, whether it be linear, quadratic, exponential, etc.
- The coordinate plane helps track how the original function \( f(x) = 3(0.75)^x - 1 \) declines and how its reflection \(-3(0.75)^x + 1\) behaves.
- By plotting these functions, we see the exponential decay graph approaching the horizontal line \( y = -1 \) and the reflected graph diverging from \( y = 1 \).
- Understanding how to use the coordinate plane is crucial for effectively visualizing and analyzing mathematical functions and their transformations.
