Question

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=\left(\frac{1}{4}\right)^x, g(x)=\left(\frac{1}{4}\right)^{x-3}+12\)

Short answer

The transformation of \(f\) represented by \(g\) includes a horizontal shift to the right by 3 units and an upwards shift by 12 units. The graph of \(f\) is an exponential decay curve originating from the y-axis, and the graph of \(g\) is the same decay curve shifted right and upwards to start at point \((3, 12)\).

Step-by-step solution

Understanding the Transformation

Function \(g\) is a transformation of function \(f\) where the exponent of the base \((\frac{1}{4})\) in function \(f\) is shifted by 3 units in function \(g\). The \(+12\) at the end of function \(g\) indicates a vertical shift upwards by 12 units. Therefore, from function \(f\) to function \(g\), there is a movement to the right by 3 units and upwards by 12 units.

Generating the Graph of Function \(f\)

Function \(f\) is an exponential function with base \((\frac{1}{4})\). Therefore, the graph of \(f\) is a decay curve starting from the y-axis and decreasing as \(x\) increases. Since the base is \(\frac{1}{4}\), the curve will decay faster compared to a base greater than 1.

Graphing the Transformation Function \(g\)

The graph of function \(g\) will be obtained by shifting the graph of function \(f\) to the right by 3 units and upwards by 12 units. Therefore, the starting point, which is originally on the y-axis in function \(f\) will now be at the point \((3,12)\) for function \(g\). The shape of the graph will be the same as \(f\) because their base is the same. The curve will continue to decay as \(x\) increases.

Key concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are crucial in modeling growth and decay processes, such as population increase or radioactive decay. The general form of an exponential function can be written as \( f(x) = a^x \), where \( a \) is a positive constant known as the base, and \( x \) is the exponent. In the function \( f(x) = \left(\frac{1}{4}\right)^x \), the base \( \frac{1}{4} \) indicates exponential decay. This is because the base is a fraction less than one, meaning the function value decreases as \( x \) increases. As a rule of thumb:

• If the base \( a > 1 \), the function exhibits exponential growth.
• If \( 0 < a < 1 \), it shows exponential decay.

This property is visible on its graph, where the function starts with a higher value on the y-axis and moves towards zero as \( x \) moves to the right.

Graphing Transformations

Graphing transformations involve shifting, stretching, compressing, or reflecting graphs of functions to obtain new functions. When dealing with exponential functions, understanding transformations is critical to identifying changes in the graph visually. In our scenario with functions \( f(x) = \left(\frac{1}{4}\right)^x \) and \( g(x) = \left(\frac{1}{4}\right)^{x-3} + 12 \), we focus on how \( g(x) \) differs from \( f(x) \):

• The term \( x-3 \) in the exponent of \( g(x) \) signifies a horizontal shift.
• The \( +12 \) added indicates a vertical shift.

Graphing transformations maintain the shape of the original function. Here, both functions are decay curves, and the only difference is their position in the coordinate plane based on the transformations applied. As a result, the graph of \( g(x) \) after transformations will appear as the graph of \( f(x) \) but moved to an entirely new location.

Vertical and Horizontal Shifts

Vertical and horizontal shifts are types of transformations applied to graphs of functions. They relocate the graph along the xy-plane without altering its shape or orientation. For horizontal shifts, the expression inside the function changes, like the \( x-3 \) in \( g(x) = \left(\frac{1}{4}\right)^{x-3} + 12 \). This particular example shifts the graph of the function 3 units to the right. The rule here is:

• A subtraction inside \( x-c \) translates the graph of the function \( c \) units to the right.
• An addition inside \( x+c \) moves it \( c \) units to the left.

Vertical shifts modify the function by adding or subtracting a constant. In \( g(x) \), the \( +12 \) moves the entire graph vertically up by 12 units. Generally, with vertical shifts:

• Addition outside the function, \( f(x) + k \), moves the graph up by \( k \) units.
• Subtraction outside, \( f(x) - k \), shifts it down by \( k \) units.

This kind of transformation helps in relocating the graph to better fit scenarios modeled by the function, without modifying the original shape.