Problem 40 - College Algebra

Question

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\left(\frac{1}{2}\right)^{x-1}+2\)

Step-by-Step Solution

Answer

The graph of the function \(f(x)\) begins at the top left, passing through (0,1), and keeps decreasing without crossing the x-axis. The graph of the function \(g(x)\) begins above the top left corner of the graph, passing through (0,3), and follows a similar curve to \(f(x)\), but one unit to the right and two units up across the whole graph. It is advisable to use a graphing utility to confirm these results.

Step 1: Identify the Shape and Basic Graph of the Functions

First, note that both functions are of the form \(f(x) = a^x\), which is the formula for an exponential function. The base of the exponential in each function is \(1/2\) which is a positive number different from 1, so these are indeed exponential functions. As the base is less than 1, these functions are decreasing exponentially. The basic graph of \(f(x) = a^x\) is a curve that starts from the top left (approaching but never reaching the y-axis), crossing through the point (0,1) and continues to decrease but never crosses the x-axis. The shape remains the same for each function but their location may differ.

Step 2: Determine Key Points and Transformations

Both functions \(f(x) = (1/2)^x\) and \(g(x) = (1/2)^{x-1}+2\) have key points and transformations. For \(f(x)\), it crosses the y-axis at the point (0,1). For \(g(x)\), it is the function \(f(x)\) shifted to the right by 1 unit and up by 2 units. So, it crosses the y-axis at the point (0,3).

Step 3: Graph the Functions

Plot the functions on the same graph using the determined shapes and key points. Make sure to clearly mark the key points and draw the curves. For \(f(x)\), it starts from the top left, passing through (0,1), and keeps decreasing without crossing the x-axis. For \(g(x)\), it starts from somewhere above the top left corner of the graph (since it's shifted upwards), passing through the point (0,3), and then follows a similar curve to \(f(x)\), but one unit to the right and two units up across the whole graph.

Key Concepts

Graphing Functions

Graphing functions is a vital skill in mathematics, helping to visualize mathematical relationships and behaviors. When graphing functions like \[f(x) = \left(\frac{1}{2}\right)^x\] and \[g(x) = \left(\frac{1}{2}\right)^{x-1} + 2,\] we often begin with sketching them on the same rectangular coordinate system. Start by identifying key points, like where each graph crosses the axes. This helps in plotting the function more precisely.
Remember that graphing functions involves looking at the overall shape. With exponential functions, the curve often approaches but never actually touches the x-axis, illustrating its behavior as x tends towards infinity or negative infinity.

Coordinate System

In mathematics, a coordinate system is a method for graphically representing points within a defined space, typically using two or three numerical values called coordinates. The most common is the rectangular (or Cartesian) coordinate system, which consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
Each point in the coordinate system is identified by an ordered pair \((x, y)\). When graphing functions, like exponential functions, understanding the coordinate plane is crucial. It enables you to determine where the graph should be plotted, ensuring that key features are accurately represented.

• Origin: The point \((0,0)\) where the axes intersect.
• Quadrants: The x and y-axis divide the plane into four areas called quadrants.
• Scales: Evenly spaced intervals on the axes, which help to size and scale your graph.

In our example, using the coordinate system allows for accurately placing the functions and observing their transformations.

Transformation of Functions

Transformations are changes made to a function's graph, altering its position and shape in the coordinate system without changing its fundamental characteristics. For exponential functions, these transformations can include translations, reflections, dilations, and rotations.
For the functions in this exercise, the key transformation involves shifting the graph. The function \[g(x) = \left(\frac{1}{2}\right)^{x-1} + 2\] is derived from \[f(x) = \left(\frac{1}{2}\right)^x\] by shifting it 1 unit to the right and 2 units upwards.

• Horizontal Shifts: Obtained by changing the input variable \((x)\) to \((x - h)\), resulting in a shift in the opposite direction of the sign of h.
• Vertical Shifts: Achieved by adding or subtracting to the function \((+k)\), shifting it upwards or downwards.

Recognizing these transformations helps with predicting the graph's position and interpreting real-world changes modeled by these functions.

Decreasing Exponential Functions

Decreasing exponential functions are a specific type of exponential function where the base is between 0 and 1. The general form is \(a^x\), where \(0 < a < 1\). As the input value increases, the function value decreases, giving the graph a distinct downward-curving appearance.
In our example with \[f(x) = \left(\frac{1}{2}\right)^x,\] the base is \(\frac{1}{2},\) which means the function is decreasing. As x becomes larger, \(\left(\frac{1}{2}\right)^x\) becomes smaller, approaching zero but never quite reaching it. This behavior is crucial in applications like radioactive decay, where quantities decrease over time.

• Key Characteristic: The function rapidly decreases but never hits the x-axis.
• Reflection Across Axes: Every decrease by a whole number value halves the function output value.

Understanding decreasing exponential functions helps in predicting long-term outcomes in scenarios involving continuous decay or reduction.