Problem 30
Question
Use the graph of \(f\) to describe the transformation that yields the graph of \(g .\) Then sketch the graphs of \(f\) and \(g\) by hand. $$f(x)=\left(\frac{1}{2}\right)^{x}, \quad g(x)=\left(\frac{1}{2}\right)^{-(x+4)}$$
Step-by-Step Solution
Verified Answer
The transformation from \(f\) to \(g\) involves a reflection across the y-axis and a horizontal shift of 4 units to the left. The graph of \(f\) is a standard decreasing exponential function while the graph of \(g\) is an increasing exponential function that has been shifted 4 units left.
1Step 1: Analyzing the functions
The given functions are \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\left(\frac{1}{2}\right)^{-(x+4)}\). For \(f\), we see that its a standard exponential function with the base less than 1—that is, it is a decreasing exponential function. For \(g\), note that the expression inside the exponent is negated, which signifies a reflection across the y-axis. Furthermore, there is an addition of 4 inside the negation, which indicates a horizontal shift.
2Step 2: Transformation from \(f\) to \(g\)
From the observation in Step 1, we can say that \(g\) is the transformation of \(f\) in two ways: (i) Because of the negation, \(f\) has been reflected across the y-axis. (ii) The addition of 4, inside the negation, causes \(f\) to shift 4 units to the left.
3Step 3: Sketching the graphs
To sketch the graphs of \(f\) and \(g\), start with \(f(x)=\left(\frac{1}{2}\right)^{x}\) which is a decreasing exponential function so it will be a graph that decreases from a high value at the negative x-axis and approaches zero at the positive x-axis. For \(g(x)=\left(\frac{1}{2}\right)^{-(x+4)}\), reflect the graph of \(f\) across the y-axis and then shift it 4 units to the left. It will still approach zero but now from the negative x-axis after the 4 units shift, and it will rise towards a higher value at the positive x-axis
Key Concepts
Graphing Exponential FunctionsHorizontal ShiftReflection Across the Y-AxisExponential Decay
Graphing Exponential Functions
Graphing exponential functions is a fundamental skill in mathematics, especially when exploring growth and decay models. An exponential function is of the form \( f(x) = a^x \) where a is a positive constant. What's key to understand is that the base a determines whether the function grows or decays as x increases.
For a greater than 1, the function represents exponential growth, increasing dramatically as x moves to the right on a graph. Conversely, when a is between 0 and 1, such as \( \frac{1}{2} \), the function showcases exponential decay, decreasing as x grows, yet never quite reaching zero - this is known as 'asymptotic' behavior. Visualizing this on a graph is crucial; the curve of an exponential decay function starts high when x is negative and approaches the x-axis (asymptote) infinitely as x becomes positive.
For a greater than 1, the function represents exponential growth, increasing dramatically as x moves to the right on a graph. Conversely, when a is between 0 and 1, such as \( \frac{1}{2} \), the function showcases exponential decay, decreasing as x grows, yet never quite reaching zero - this is known as 'asymptotic' behavior. Visualizing this on a graph is crucial; the curve of an exponential decay function starts high when x is negative and approaches the x-axis (asymptote) infinitely as x becomes positive.
Horizontal Shift
In the world of function transformations, a 'horizontal shift' is a type of shift along the x-axis. This transformation is simple to recognize; it occurs when we add or subtract a constant to the input variable x.
For illustration, consider the function \( f(x) = a^(x-h) \) where h is the horizontal shift. If h is positive, the entire graph of f moves h units to the right. If h is negative, it shifts h units to the left. In the current exercise, the function g introduces a shift of 4 units to the left, because it essentially involves a regression of x by 4 before applying the exponential operation.
For illustration, consider the function \( f(x) = a^(x-h) \) where h is the horizontal shift. If h is positive, the entire graph of f moves h units to the right. If h is negative, it shifts h units to the left. In the current exercise, the function g introduces a shift of 4 units to the left, because it essentially involves a regression of x by 4 before applying the exponential operation.
Reflection Across the Y-Axis
Reflection across the y-axis is another transformation that can dramatically change the appearance of a graph. It is akin to flipping the function over the y-axis, thus swapping the left and right sides of the graph.
In mathematical terms, a reflection is noted by a negative sign in front of the input variable x. So, a function written as \( f(x) = a^{-x} \) is a reflection of \( f(x) = a^{x} \) across the y-axis. For example, in the exercise provided, the function g reveals a reflection due to the negative sign before the x in the exponent. This transformation implies that instead of the graph moving down to the right, like a typical exponential decay, it descends to the left.
In mathematical terms, a reflection is noted by a negative sign in front of the input variable x. So, a function written as \( f(x) = a^{-x} \) is a reflection of \( f(x) = a^{x} \) across the y-axis. For example, in the exercise provided, the function g reveals a reflection due to the negative sign before the x in the exponent. This transformation implies that instead of the graph moving down to the right, like a typical exponential decay, it descends to the left.
Exponential Decay
Exponential decay is a specific type of exponential function where the value decreases over time. It is characterized by the function \( f(x) = a^x \) with 0 < a < 1.
An exponential decay curve depicts a rapid decrease at first, which slowly tapers off. The classic real-world examples of exponential decay include radioactive decay and cooling of hot objects. In the context of the exercise, the function f illustrates this decay with the base \( \frac{1}{2} \); it decreases swiftly as x becomes less negative, ultimately flattening as it approaches the x-axis. This concept is vital to understand as it represents how certain processes diminish over time in a predictable and quantifiable fashion.
An exponential decay curve depicts a rapid decrease at first, which slowly tapers off. The classic real-world examples of exponential decay include radioactive decay and cooling of hot objects. In the context of the exercise, the function f illustrates this decay with the base \( \frac{1}{2} \); it decreases swiftly as x becomes less negative, ultimately flattening as it approaches the x-axis. This concept is vital to understand as it represents how certain processes diminish over time in a predictable and quantifiable fashion.
Other exercises in this chapter
Problem 30
Solve the exponential equation. $$\left(\frac{3}{4}\right)^{x}=\frac{64}{27}$$
View solution Problem 30
Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. (Value) $$x=345$$ $$x=\frac{4}{5}$$ $$x=1
View solution Problem 31
The table shows the yearly sales \(S\) (in millions of dollars) of Whole Foods Market for the years 2006 through 2013. (Source: Whole Foods Market) $$\begin{arr
View solution Problem 31
Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{1 / 2}(x-2)$$ .
View solution