Problem 13 - Contemporary Precalculus

Question

List the transformations needed to transform the graph of \(h(x)=2^{x}\) into the graph of the given function. $$k(x)=3\left(2^{x}\right)$$

Step-by-Step Solution

Answer

Answer: The transformation from the base function \(h(x) = 2^x\) to the given function \(k(x) = 3\left(2^x\right)\) is a vertical scaling by a factor of 3.

Step 1: Identify the base function and the given function

The base function is \(h(x) = 2^x\), and the given function is \(k(x) = 3\left(2^x\right)\). Step 2:

Step 2: Compare the two functions

Notice that the given function \(k(x)\) is the base function \(h(x)\) multiplied by a constant (3). The base function \(h(x)\) represents an exponential function, and \(k(x)\) can be considered a transformation (scaling) of the base function. Step 3:

Step 3: Identify the transformation

Observe that the given function \(k(x) = 3\left(2^x\right)\) is equal to \(h(x)\) multiplied by 3. This is a vertical scaling, which means that the graph of \(k(x)\) is a vertical stretch of the graph of \(h(x)\) by a factor of 3. The function \(k(x)\) is created by multiplying every \(y\)-value of \(h(x)\) by 3. To summarize, there is only one transformation from the base function \(h(x) = 2^x\) to the given function \(k(x) = 3\left(2^x\right)\), which is a vertical scaling of 3.

Key Concepts

Vertical Scaling

Vertical scaling is a type of graph transformation that involves stretching or compressing the graph of a function along the vertical axis. In the context of exponential functions, vertical scaling changes the output values of the function without affecting the input values.

In the exercise presented, we have the function \(h(x) = 2^x\) as the base function. When this function is multiplied by a constant, such as 3 in the given function \(k(x) = 3 \cdot 2^x\), every output or \(y\)-value is scaled by this multiplier (3). This means:

• The points on the graph of \(h(x)\) are pushed further away from the x-axis if the scaling factor is greater than 1, as is the case here.
• If the multiplier was a fraction between 0 and 1, the graph would be compressed toward the x-axis instead.

Vertical scaling can significantly impact the visual representation of a function, showcasing the power of transformation in making mathematical expressions visually dynamic.

Exponential Function

An exponential function is a mathematical function in the form \(f(x) = a^x\), where \(a\) is a constant and \(x\) is the exponent variable. These functions are characterized by their rapid growth or decay, depending on the base \(a\).

In the given exercise, the base function \(h(x) = 2^x\) is a typical example where the base \(2\) is greater than 1, resulting in a graph that increases exponentially.

• Exponential growth occurs when the base is greater than 1, leading to rapid increases as \(x\) becomes larger.
• Conversely, if the base is between 0 and 1, the function describes exponential decay.

The exponential function has a unique graph that passes through the point \((0,1)\) because any non-zero number raised to the 0 power is 1. It also has a horizontal asymptote on the x-axis, as the function never quite reaches zero or negative values but approaches them as \(x\) heads towards negative infinity.

Graph Transformation

Graph transformations modify the position or shape of a function's graph relative to its original position. These include translation, reflection, and scaling—like vertical scaling—in various axes.

In our exercise, the transformation from \(h(x) = 2^x\) to \(k(x) = 3 \cdot 2^x\) is solely a vertical scaling. Unlike other transformations that might shift or mirror the graph, vertical scaling maintains the overall form of the graph but changes its height.

• Transformations preserve the x-values but modify the y-values as dictated by transformation rules.
• Aside from vertical scaling, transformations could also occur horizontally, for example by changing the base of the exponent.

Understanding these transformations helps students and mathematicians alike to visualize and manipulate functions to fit data or solve applied problems. Each transformation provides a different lens through which the behavior of mathematical functions can be examined, altered, and applied to real-world situations.