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Chapter 5: Problem 19

In Exercises \(15-20\) , sketch the graph of the function. $$h(x)=5^{x-2}$$

Short Answer

Expert verified
The graph of \(h(x) = 5^{x-2}\) starts from Y-intercept (0, 0.04) and rises to the right, it does not have any real x-intercept. The function is defined for all real number x, and as x increases, \(h(x)\) also increases.

Step by step solution

01

Understanding the Base Function

Firstly, understand the base function \(f(x) = 5^x\). This is an exponential function where the base, 5, is a constant greater than 1. The function is defined for all real numbers. As x approaches negative infinity, \(5^x\) approaches zero and as x approaches positive infinity, \(5^x\) also approaches positive infinity.
02

Transformation

The function \(h(x) = 5^{x-2}\) is a transformation of the base function. The '-2' inside the exponent means the graph of \(f(x) = 5^x\) is shifted 2 units to the right. Hence, the equation for the graph becomes \(y = 5^{x-2}\).
03

Finding the intercepts

For the Y-intercept, set x to 0, hence \(h(0) = 5^{0-2} = 5^{-2} = 0.04\). Therefore the y-intercept is at (0,0.04). For the x-intercept, \(h(x) = 0\), but since \(h(x)\) is an exponential function and \(5^{x-2} > 0\) for all real values of x, the graph does not cross the x-axis hence has no x-intercept.
04

Sketch the graph

Plot the y-intercept on the graph, from step 2, we know that the graph is shifted 2 units to the right from the standard \(5^x\) graph and from step 3, we don't have any x-intercept. Plot a few more points if necessary and draw a smooth curve connecting the points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Transformations
When dealing with graph transformations, we are essentially shifting, stretching, or compressing a base graph to obtain a new one. The function given in the exercise is \(h(x) = 5^{x-2}\), which is a transformation of the base function \(f(x) = 5^x\).
The exponent modification \(x-2\) indicates a horizontal shift. Here鈥檚 how it works:
  • A transformation such as \(f(x) = 5^{x-a}\) shifts the graph \(a\) units to the right if \(a\) is positive.
  • If \(a\) were negative, it would shift the graph to the left.
  • Therefore, \(5^{x-2}\) moves the graph of \(5^x\) two units to the right on the x-axis.

      • This type of transformation does not affect the shape of the graph; it only repositions it. This same method of shifting applies to many types of functions, not just exponential ones.
Exponential Growth
Exponential growth refers to increasingly rapid growth as the value of \(x\) in an exponential function increases. In the context of our function \(h(x) = 5^{x-2}\), the base of the exponent is 5, which is greater than 1. This tells us that the function represents exponential growth.
Here are a few key characteristics of exponential growth:
  • As \(x\) increases, \(h(x)\) rises sharply and continually at an ever-increasing rate.
  • The graph will have a horizontal asymptote, typically at \(y=0\), indicating that as \(x\) moves toward negative infinity, \(h(x)\) approaches zero but never actually touches the x-axis.
  • Because of this rapid increase, exponential functions are often used to model scenarios like population growth and interest compounding.

Understanding this concept is crucial for interpreting real-world models that utilize exponential functions, allowing predictions about future growth based on current data.
Function Intercepts
Function intercepts help us find where the graph of a function intersects the axes. In our example, \(h(x) = 5^{x-2}\), we are particularly interested in finding the y-intercept and any possible x-intercepts to better understand our graph.
**Y-Intercept**:
  • Set \(x = 0\) to find the y-intercept: \(h(0) = 5^{0-2} = 5^{-2} = 0.04\).
  • This means the graph crosses the y-axis at \((0, 0.04)\).

**X-Intercept**:
  • The x-intercept requires solving \(h(x) = 0\), but since \(5^{x-2} > 0\) for any real \(x\), \(h(x)\) can never be zero.
  • This means that there are no x-intercepts for this function.

These intercepts help plot the function more accurately by giving critical points where the graph intersects the axes.

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