Problem 26 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. Select integers from \(-2\) to 2 , inclusive, for \(x\). Then describe how the graph of g is related to the graph of \(f .\) If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=2^{x} \quad \text { and } \quad g(x)=2^{x+2}$$

Step-by-Step Solution

Answer

The graph of \(g(x)=2^{x+2}\) is identical to the graph of \(f(x)=2^{x}\), but shifted 2 units to the left.

Step 1: Graph of function \(f(x) = 2^x\)

To start, select integers from -2 to 2. For these values, calculate the corresponding \(f(x)\) and plot them on the graph.\n\nFor instance:\n\n\(f(-2) = 2^{-2} = 0.25\)\n\n\(f(-1) = 2^{-1} = 0.5\)\n\n\(f(0) = 2^{0} = 1\)\n\n\(f(1) = 2^{1} = 2\)\n\n\(f(2) = 2^{2} = 4\)

Step 2: Graph of function \(g(x) = 2^{x+2}\)

Similarly, for \(g(x)\), select the same integers from -2 to 2, compute \(g(x)\), and plot these on the graph.\n\nFor example:\n\n\(g(-2) = 2^{0} = 1\)\n\n\(g(-1) = 2^{1} = 2\)\n\n\(g(0) = 2^{2} = 4\)\n\n\(g(1) = 2^{3} = 8\)\n\n\(g(2) = 2^{4} = 16\)

Step 3: Analyze and Compare both functions

Once both functions are graphed, observe the relationships between \(f(x)\) and \(g(x)\).\n\nYou can now see how the graph of \(g(x)\) is a shift of the graph of \(f(x)\). More specifically, \(g(x) = 2^{x+2}\) is a shift 2 units to the left from the graph of \(f(x) = 2^{x}\). This fact can be confirmed by comparing the corresponding values of \(x\) in both functions.

Key Concepts

Coordinate System

Understanding the coordinate system is the foundation of graphing any function. It consists of two perpendicular lines, commonly called axes, which intersect at a point known as the origin. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. Together, they divide the plane into four quadrants. Numbers to the right of the origin on the x-axis are positive, while those to the left are negative. Similarly, numbers above the origin on the y-axis are positive, and those below are negative.

When graphing functions, you typically plot a set of points, where each point represents a pair of an input value and its corresponding output value from the function. For the exercise with functions f(x) = 2^x and g(x) = 2^{x+2}, we choose integer values for x and find the respective y values. By plotting these points for both functions in the same coordinate system, we can visually compare their behaviors and how one relates to the other.

Function Transformation

Function transformation involves altering the original function's graph in some way. Common transformations include shifting, stretching, compressing, and reflecting. When you look at the functions f(x) = 2^x and g(x) = 2^{x+2}, you are essentially viewing a horizontal shift.

For each value of x, adding 2 inside the function—before applying the exponential—shifts all the points of the graph to the left by 2 units. This is because the function value that you used to get for x is now obtained for x + 2. Graphing f(x) and g(x) will visually demonstrate this shift. Students can confirm this by noting that the point which was at (0, 1) on f(x) is now at (-2, 1) on g(x). Recognizing this pattern is crucial for understanding transformations without the need to calculate every single point.

Exponential Growth

Exponential growth is a process that increases quantity over time. In mathematics, an exponential function is a powerful tool to represent this type of growth. Its general form is f(x) = a^x, where a is a positive constant called the base and x is the exponent. The graph of an exponential function is a curve that starts slowly and increases more and more rapidly as x increases.

With the functions given in the exercise, f(x) = 2^x shows exponential growth with a base of 2. As x increases by 1, the value of f(x) doubles, which is a hallmark of exponential growth. For the given range of integers from -2 to 2, the value of f(x) grows from 0.25 to 4. This growth can be visualized on the graph as a curve rising steeply as we move to the right on the x-axis. Understanding exponential growth is not only key in mathematics but also in real-world applications like population growth, finance, and natural sciences, where processes grow at rates proportional to their current value.