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Chapter 6: Problem 7

The graph of \(f(x)=2\left(\frac{1}{4}\right)^{x-20}\) is shifted downward 4 units, and then shifted left 2 units, stretched vertically by a factor of 4 , and refl cted about the \(x\) -axis. What is the equation of the new function, \(g(x) ?\) State its \(y\) -intercept, domain, and range.

Short Answer

Expert verified
The new function is \( g(x) = -8\left(\frac{1}{4}\right)^{x-18} + 16 \). The \( y \)-intercept is \( -8\cdot 4^{18} + 16 \), the domain is \( (-\infty, \infty) \), and the range is \( (-\infty, 16) \).

Step by step solution

01

Identify the Original Function

The given function is \( f(x) = 2\left(\frac{1}{4}\right)^{x-20} \). This is an exponential function with a base of \( \frac{1}{4} \) that has been translated right by 20 units and vertically stretched by a factor of 2.
02

Apply Downward Shift

To shift down by 4 units, subtract 4 from the function: \( h(x) = 2\left(\frac{1}{4}\right)^{x-20} - 4 \).
03

Apply Horizontal Shift

Now, shift the function left by 2 units. Adjust the inputs: \( k(x) = 2\left(\frac{1}{4}\right)^{(x+2)-20} - 4 = 2\left(\frac{1}{4}\right)^{x-18} - 4 \).
04

Apply Vertical Stretch

Stretch the function vertically by a factor of 4. Multiply the entire function by 4: \( m(x) = 4 \cdot \left( 2\left(\frac{1}{4}\right)^{x-18} - 4 \right) = 8\left(\frac{1}{4}\right)^{x-18} - 16 \).
05

Apply Reflection

Reflect the function across the \( x \)-axis by negating the entire function: \( g(x) = -(8\left(\frac{1}{4}\right)^{x-18} - 16) = -8\left(\frac{1}{4}\right)^{x-18} + 16 \).
06

Determine the y-intercept

The \( y \)-intercept occurs when \( x = 0 \). Evaluate \( g(0) = -8\left(\frac{1}{4}\right)^{-18} + 16 \). Simplifying, \( g(0) = -8\cdot 4^{18} + 16 \).
07

Determine the Domain

Since the function is exponential, its domain is all real numbers, i.e., \( (-\infty, \infty) \).
08

Determine the Range

The range of \( g(x) \) is determined by the reflection and vertical transformations applied. Since it was originally an exponential decay starting at \( y = 2 \), and reflected about the \( x \)-axis, the range is \( (-\infty, 16) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are a special type of function where the variable, typically denoted as \( x \), appears in the exponent. These functions are incredibly useful in modeling a variety of real-world phenomena, such as population growth and radioactive decay.
In the general form, an exponential function can be written as \( f(x) = a \, b^x \) where:
  • \( a \) is a constant that affects the vertical stretch or compression of the graph,
  • \( b \) is the base of the power. If \( b > 1 \), the function shows exponential growth. If \( 0 < b < 1 \), it is exponential decay.
In the exercise, the original function \( f(x)=2\left(\frac{1}{4}\right)^{x-20} \) represents an exponential decay because the base is \( \frac{1}{4} \), which is less than one. This means as \( x \) increases, the function value decreases, causing the graph to slope downward.
Graph Transformations
Graph transformations refer to the various ways you can manipulate a function to produce a different graph.
Each type of transformation modifies the function's appearance in specific ways.
In this exercise, multiple transformations are applied:
  • Vertical Shift: Shifting the graph up or down by adding or subtracting from the function. For the exercise, subtracting 4 moved the graph downward by 4 units.
  • Horizontal Shift: Moving the graph left or right involves changing the input value of the function. Here, substituting \((x + 2)\) for \(x\) shifted the function to the left by 2 units.
  • Vertical Stretch: This transformation involves multiplying the function by a constant greater than 1, making the graph taller. Multiplying the entire function by 4 stretched it vertically in the exercise.
These transformations alter the position and shape of the graph, allowing us to tailor the function to specific data or desired characteristics.
Function Reflection
Function reflection is a transformation that flips the graph of a function across an axis.
In mathematical terms, reflecting a graph across the x-axis involves negating the entire function: \( g(x) = -f(x) \). This turns all values of the function from positive to negative and vice versa.
For the given problem, after performing other transformations, reflecting the function across the x-axis resulted in the function \( g(x) = -8\left(\frac{1}{4}\right)^{x-18} + 16 \). This makes the graph appear as a mirror image across the x-axis and inverses the direction of the graph's slope. The previous decreasing pattern now appears to increase downward when viewed as a reflection.
Function Domain and Range
In mathematics, the domain and range of a function are pivotal in understanding the "boundaries" of its behavior and graph.
  • Domain: This is the set of all possible \( x \) values (inputs) that the function can accept. For exponential functions, you can plug in any real number, giving it a domain of \( (-\infty, \infty) \).
  • Range: The range is the set of possible \( y \) values (outputs) a function can produce. In this exercise, since the function has been transformed, including a reflection which changed how the graph presents its output values, the range of the reflected function becomes \((-\infty, 16) \).
Understanding domain and range helps us predict what the function outputs might be for given inputs, and where its graph might lie on the coordinate plane.

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