Problem 26 - Algebra and Trigonometry

Question

begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ g(x)=2^{x+2} $$

Step-by-Step Solution

Answer

The graph of \(f(x)=2^{x}\) has a y-intercept at (0, 1) and a horizontal asymptote at \(y=0\). Its domain is all real numbers, and its range is all positive real numbers. The graph of \(g(x)=2^{x+2}\) is identical to \(f(x)=2^{x}\), but is shifted two units to the left. It has a y-intercept at (0, 4) and a horizontal asymptote at \(y=0\). Its domain is all real numbers, and its range is all positive real numbers.

Step 1: Graph \(f(x)=2^{x}\)

The graph of \(f(x)=2^{x}\) is the base graph of an exponential function. This function has a y-intercept of 1 (since \(f(0)=1\) ) and an asymptote at \(y=0\). The domain of \(f(x)=2^{x}\) is all real numbers, and the range is all positive real numbers.

Step 2: Transform the graph to obtain \(g(x)=2^{x+2}\)

The function \(g(x)=2^{x+2}\) is equal to \(f(x+2)\), which means there is a horizontal shift to the left by 2 units from \(f(x)=2^{x}\). Therefore, the graph is identical to \(f(x)=2^{x}\), but moved two units to the left.

Step 3: Analyze the graph of \(g(x)=2^{x+2}\)

Because the base graph just shifted horizontally, the overall shape hasn't changed. Thus, the horizontal asymptote remains \(y=0\), and the y-intercept is the point (0, 4) (since \(g(0)=4\)). The domain of \(g(x)=2^{x+2}\) is all real numbers, and the range is all positive real numbers.

Step 4: Verify the results

If possible, utilize a graphing utility/tool to confirm the hand-drawn graphs and to cross-check the answers. The accuracy of the drawn graphs, domain, range and asymptotes can be compared with the utility's output.

Key Concepts

Graph Transformations

Graph transformations are adjustments made to the original graph of a function. In this case, we start with the graph of the basic exponential function, \(f(x) = 2^{x}\). The transformation involves shifting, reflecting, or scaling the graph. For \(g(x) = 2^{x+2}\), we perform a horizontal shift.

• The expression \(2^{x+2}\) indicates a horizontal shift.
• The \(+2\) inside the function's exponent shifts the graph 2 units to the left.

The fundamental shape of the graph does not change. The curve still rises sharply as \(x\) becomes larger and approaches zero as \(x\) decreases. Understanding how transformations affect the position and shape of graphs helps in graphing the functions accurately without plotting numerous points.

Domain and Range

The domain and range of a function tell us about the permissible inputs (x-values) and possible outputs (y-values) of the function respectively. For exponential functions like \(f(x) = 2^x\) and \(g(x) = 2^{x+2}\), understanding these concepts is crucial.

• **Domain:** For both functions, the domain is all real numbers, \((-\infty, \infty)\). You can input any real number from negative to positive infinity.
• **Range:** For exponential functions growing exponentially upwards, like these, the range is all positive real numbers, \((0, \infty)\).

This means the function never touches or crosses the \(x\)-axis, except when considering transformations like reflections across axes.

Asymptotes

An asymptote is a line that a graph approaches but never actually touches. Exponential functions generally have horizontal asymptotes.

• The base function \(f(x) = 2^x\) has a horizontal asymptote at \(y=0\).
• With the transformed function \(g(x) = 2^{x+2}\), the horizontal asymptote remains at \(y=0\) since only a horizontal shift is applied.

The graph of the function gets infinitely close to the asymptote, but it won't intersect it. This property is essential in predicting and understanding the behavior of exponential functions.