Problem 51 - Algebra for College Students

Question

Graph \(f(x)=2^{x}\). Where should the graphs of \(f(x)=\) \(2^{x-5}, f(x)=2^{x-7}\), and \(f(x)=2^{x+5}\) be located? Graph all three functions on the same set of axes with \(f(x)=2^{x}\).

Step-by-Step Solution

Answer

The graph of \(f(x)=2^{x-5}\) shifts right by 5 units, \(f(x)=2^{x-7}\) right 7 units, and \(f(x)=2^{x+5}\) left 5 units.

Step 1: Graph the Base Function

The base function is given as \(f(x) = 2^x\). This is an exponential function with a base of 2, which means it grows exponentially as \(x\) increases. The graph of this function passes through the point (0,1) because \(2^0 = 1\) and it increases steeply as \(x\) becomes larger.

Step 2: Graph the Function with Horizontal Translation to the Right

Consider \(f(x) = 2^{x-5}\). This represents a horizontal translation of the original function to the right by 5 units. Essentially, at any given \(y\)-value, \(x\) is 5 units larger than it would be for \(f(x)=2^x\). Graphically, this means the entire graph of \(f(x)=2^x\) is shifted to the right by 5 units.

Step 3: Graph the Function with a Greater Horizontal Translation to the Right

For \(f(x) = 2^{x-7}\), the graph is horizontally translated further to the right by 7 units. This is similar to the previous step but results in an additional right shift of 2 units beyond \( f(x) = 2^{x-5} \). On the graph, it means taking the graph of \(f(x) = 2^x\) and moving it 7 units to the right.

Step 4: Graph the Function with Horizontal Translation to the Left

Now consider \(f(x) = 2^{x+5}\). Here, the graph is horizontally translated to the left by 5 units. This is because each \(y\)-value is achieved 5 units earlier (in terms of \(x\)) than it would be for \(f(x) = 2^x\). On the graph, this means the entire curve shifts to the left by 5 units.

Step 5: Visual Representation on the Same Axes

To represent these functions on the same set of axes, begin by plotting \(f(x)=2^x\). Then apply the transformations visually: Shift the curve for \(f(x)=2^x\) to the right by 5 units for \(f(x)=2^{x-5}\), to the right by 7 units for \(f(x)=2^{x-7}\), and to the left by 5 units for \(f(x)=2^{x+5}\). This visualization helps in understanding how the original function changes with different translations.

Key Concepts

Exponential Growth

Exponential growth is a fascinating concept because it describes how quantities can increase rapidly over time. In mathematics, exponential growth occurs in functions of the form \( f(x) = a^x \), where \( a \) is a positive constant greater than 1. The function \( f(x) = 2^x \) is a classic example of exponential growth. This function grows rapidly as \( x \) increases.
One of the key characteristics of exponential functions is that the rate of growth itself increases. This means each subsequent value is obtained by multiplying the previous value by a constant factor. For example, if you graph the function \( f(x) = 2^x \), you will notice that it passes through the point (0, 1), since \( 2^0 = 1 \). It rapidly shoots upwards as \( x \) moves to the right, illustrating how small increases in \( x \) result in large increases in \( f(x) \).

• Exponential growth functions grow quicker than linear or quadratic functions over time.
• These functions are often used to model real-world phenomena, such as population growth, radioactive decay, and interest in finance.

This understanding helps when graphing these functions as you'll anticipate rapid increases within short intervals along the \( x \)-axis.

Horizontal Translation

A horizontal translation in the context of graphing functions refers to shifting the entire graph left or right. For the function \( f(x) = 2^{x} \), horizontal translations can be expressed in the form \( f(x) = 2^{x-h} \), where the value \( h \) determines the direction and magnitude of the shift.
When considering \( f(x) = 2^{x-5} \), you are performing a horizontal shift to the right by 5 units. Conversely, for \( f(x) = 2^{x+5} \), the graph shifts to the left by 5 units. These changes mean that each \( x \)-value on the original graph corresponds to \( x+h \) on the translated graph for \( x-h \). Graphically, this does not affect the shape or steepness of the graph, but it changes its position.

• Shifts to the right: \( f(x) = 2^{x-h} \), \( h > 0 \)
• Shifts to the left: \( f(x) = 2^{x+h} \), \( h > 0 \)

Understanding this helps to quickly reposition graphs and analyze dependencies of the function's output on input values.

Function Transformation

Function transformations include any alteration to the original form of a function. This can involve translations, reflections, rotations, or rescaling of graphs. In this context, we are focusing specifically on horizontal translations of the exponential function \( f(x) = 2^x \). Transforming a function modifies its graphical representation to help highlight certain behaviors or adapt to real-world data.
For example, when you graph the transformed function \( f(x) = 2^{x-7} \), you are shifting the base function \( f(x) = 2^x \) to the right by 7 units. This type of transformation is a strategic modification to alter the input that produces desirable alterations in the output. The combined transformations seen in the example — such as shifting both to the left and right — illustrate the robust flexibility such manipulation provides.

• Horizontal shifts: Repositioning along the \( x \)-axis while maintaining the same overall shape.
• Practically applies to align mathematical models with observed data or to simplify graph interpretation.

Through these transformations, grasping complex functions and linking them to real scenarios becomes much more manageable.