- Analyze the functions: \(f(x) = 7^x\) is a standard exponential function with a horizontal asymptote at \(y = 0\).
- Identify the transformation: \(g(x) = 7^x + 6\) represents a vertical upward shift of the function \(f(x)\) by 6 units.
- Determine the new asymptote: Shifting the horizontal asymptote \(y = 0\) upward by 6 units results in the new horizontal asymptote \(y = 6\).
- Select the true statement: The asymptote of \(g(x)\) is the asymptote of \(f(x)\) shifted six units up. \(\boxed{\text{The asymptote of } g(x) \text{ is the asymptote of } f(x) \text{ shifted six units up.}}\)
Explanation
- Analyze the functions \(f(x)\) and \(g(x)\)
We are given two functions: \(f(x) = 7^x\) and \(g(x) = 7^x + 6\). The function \(f(x) = 7^x\) is an exponential function. The domain of any exponential function \(a^x\) (where \(a > 0, a
eq 1\)) is the set of all real numbers, \(\mathbb{R}\). The range of \(f(x) = 7^x\) is \((0, \infty)\), meaning all \(y > 0\). The horizontal asymptote of \(f(x)\) is found by considering the limit as \(x \to -\infty\), which gives \(y = 0\).
- Determine the transformation and effect on the asymptote
The function \(g(x) = 7^x + 6\) is obtained by adding a constant, 6, to \(f(x)\). According to the rules of function transformations, if a function \(h(x)\) is defined as \(h(x) = f(x) + k\), the graph of \(h(x)\) is the graph of \(f(x)\) shifted vertically by \(k\) units. If \(k > 0\), the shift is upward; if \(k < 0\), the shift is downward. In this case, \(k = 6\), which is positive. Therefore, the graph of \(g(x)\) is the graph of \(f(x)\) shifted 6 units upward.
- Evaluate the statements and conclude
Since the graph of \(g(x)\) is shifted vertically upward by 6 units, all features of the graph, including the horizontal asymptote, are also shifted upward by the same amount. The horizontal asymptote of \(f(x)\) is the line \(y = 0\). Shifting this line 6 units upward results in the new horizontal asymptote \(y = 0 + 6 = 6\). Thus, the asymptote of \(g(x)\) is the asymptote of \(f(x)\) shifted six units up. This confirms that the fourth statement provided in the options is the correct one.
Examples
Imagine you are tracking the growth of a population of bacteria in a lab. Initially, the population follows the growth model \(P(t) = 7^t\). Later, you introduce a constant influx of 6 new bacteria per hour, so the new population model becomes \(P(t) = 7^t + 6\). Understanding that adding a constant shifts the growth curve vertically helps you visualize that the entire population size at any given time \(t\) has simply increased by 6 units compared to the original model. This demonstrates how vertical translations help in adjusting models to account for constant additions or offsets in real-world data.