Problem 29 - 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. $$f(x)=2^{x} \text { and } g(x)=2^{x}+1$$

Step-by-Step Solution

Answer

The graph of \(g(x)\) is the same as the graph of \(f(x)\) but shifted one unit up.

Step 1: Plotting of Function f(x)

Begin by plotting function \(f(x) = 2^x\). Select the values -2, -1, 0, 1, and 2 for \(x\), and compute the corresponding \(y\) values. Here are the points: (-2 , \(2^{-2} = 0.25\)), (-1 , \(2^{-1} = 0.5\)), (0 , \(2^0 = 1\)), (1 , \(2^1 = 2\)), (2 , \(2^2 = 4\)). Plot these points on a graph and then sketch the curve.

Step 2: Plotting of Function g(x)

Next, plot the function \(g(x) = 2^x + 1\). As before, select the values -2, -1, 0, 1, and 2 for \(x\), and compute the corresponding \(y\) values. Here are the points: (-2 , \(2^{-2} + 1 = 1.25\)), (-1 , \(2^{-1} + 1 = 1.5\)), (0 , \(2^0 + 1 = 2\)), (1 , \(2^1 + 1 = 3\)), (2 , \(2^2 + 1 = 5\)). Plot these points on the same graph and then sketch the curve of \(g\).

Step 3: Comparison of f(x) and g(x)

Finally, describe how the graph of \(g\) is related to the graph of \(f\). You should notice that the graph of \(g(x) = 2^x + 1\) is the same as the graph of \(f(x) = 2^x\) but shifted one unit up. This is because the '+1' in \(g(x) = 2^x + 1\) adds 1 to every \(y\) value of \(f\), shifting the entire graph upwards.

Key Concepts

Exponential Functions

Exponential functions are special mathematical expressions where the variable is an exponent. Unlike linear functions that change at a constant rate, exponential functions grow or decay exponentially. This means their rate of change increases or decreases rapidly.

Considering the function given in the problem,

• We have the base, which is 2 in this case. This base indicates the type of growth or decay of the function.
• The function notation is written as \(f(x) = 2^x\).
• This shows us that for every increase in \(x\) by 1, the value of \(f(x)\) doubles.

This doubling effect is what makes exponential functions so powerful, as small changes in \(x\) lead to significant changes in \(f(x)\). Hence, they are often used in modeling various real-world phenomena like population growth or radioactive decay. Understanding the base and exponent helps determine the behavior of these functions.

Graph Transformations

Graph transformations help us understand how modifying a function affects its graph. In our exercise, we see a transformation applied to the exponential function.

When we compare \(f(x) = 2^x\) and \(g(x) = 2^x + 1\):

• The \(+1\) in \(g(x)\) indicates a vertical translation.
• Each point on the graph of \(f(x)\) is moved up by one unit to create the graph of \(g(x)\).

This vertical shift means that while the shape and growth rate of the function remain unchanged, the entire function has been elevated on the coordinate plane.

Graph transformations such as translations, reflections, and scaling allow us to adjust and manipulate the appearance of graphs without changing their fundamental form. Understanding these transformations can greatly aid in predicting the effect of modifications in various contexts.

Coordinate System

A coordinate system is an essential tool for graphing functions, like the ones in the exercise. The most common type is the Cartesian coordinate system, which uses two perpendicular axes:

• The horizontal axis is called the \(x\)-axis.
• The vertical axis is the \(y\)-axis.

Together, they create a grid where each point is expressed as an ordered pair \((x, y)\). In graphing functions, we use this system to plot points calculated using the function's rules.

For instance, for \(f(x) = 2^x\), you calculate the values, like \((1, 2)\) or \((2, 4)\), and plot these points on the graph. This visual representation allows us to clearly see the behavior and growth of the function. Having a solid grasp of the coordinate system is crucial for accurately plotting points and interpreting the resulting graphs. This understanding aids in better analysis of relationships between variables and helps us visually grasp complex concepts such as functions and transformations.