Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$f(x)=5^{-x}$$

Step-by-Step Solution

Answer

The function \(f(x) = 5^{-x}\) has a horizontal asymptote at \(y = 0\), has a y-intercept at the point (0,1), and is decreasing. The function doesn't have a x-intercept.

Step 1: Understand the Function

The function given is an exponential function of the form \(f(x) = a^x\), where a is a positive constant. In our case, \(a = 5\) and \(f(x) = 5^{-x}\), which is equivalent to \(f(x) = 1/5^x\). This tells us that as 'x' increases, the value of 'f(x)' decreases. This is characteristic of a decreasing function.

Step 2: Identify the Asymptotes

An asymptote of a function is a line that the function approaches but never touches. For any exponential function of the form \(f(x) = a^x\) or \(f(x) = 1/a^x\), the x-axis (i.e., the line \(y = 0\)) is a horizontal asymptote. This is because, no matter how large or small 'x' becomes, 'f(x)' will never equal zero.

Step 3: Identify the Intercepts

The x-intercept of a function is the value of 'x' at which the function equals zero. However, as we determined in the previous step, the function \(f(x) = 1/5^x\) never equals zero. Therefore, this function has no x-intercepts. The y-intercept of a function is the value of 'x' at which the function equals zero. Putting \(x=0\) in \(f(x) = 1/5^x\) gives us \(y=1\), therefore the function has a y-intercept at point (0,1).

Step 4: Determine the Direction of the Function

To determine whether the function is increasing or decreasing, we look at the exponent of the function. Because our exponent is \(-x\), this means that as 'x' increases, the value of the function decreases. Therefore, the function is decreasing.

Step 5: Sketch the Graph

To sketch the graph, plot the y-intercept at point (0,1), sketch the x-axis as a horizontal asymptote, and draw a curve that starts in the upper left, touches the y-intercept, and then approaches the x-axis as 'x' goes to positive infinity. The direction of the curve indicates the function is decreasing.

Key Concepts

Asymptotes

In exponential functions, asymptotes play a crucial role. An asymptote is a line that a function approaches but does not reach. For the function \(f(x) = 5^{-x}\), we consider horizontal asymptotes. As \(x\) becomes very large, \(5^{-x}\) becomes very small, moving towards zero but never quite reaching it. Hence, the x-axis, represented by the line \(y = 0\), acts as the horizontal asymptote.

• As \(x\) increases, the function \(f(x)\) decreases.
• The curve gets close to but does not touch the x-axis.
• This is typical for functions of the form \(a^{-x}\) where \(a > 0\).

Remember, asymptotes help us define the behavior of the graph far from the origin. They are not intersected by the function's curve.

Graphing Functions

When graphing exponential functions like \(f(x) = 5^{-x}\), some principles guide us. The graph of this function will have a distinctive shape, reflecting the decreasing nature of the function. Let's outline the graphing steps:

• **Identify Important Points**: Start with the y-intercept at \(x = 0\). In this case, \(f(0) = 1\), so the point is \((0, 1)\).
• **Consider Asymptotes**: Include the x-axis as a horizontal asymptote, indicating that the graph will approach but never reach \(y = 0\).
• **Sketch the Curve**: Start from the y-intercept, draw the curve towards the upper left (as \(x\) decreases), and approach the x-axis as \(x\) increases on the right. This shows the function decreasing.

Be careful to maintain the general shape, which reflects the always-decreasing nature of this function.

Intercepts

Intercepts are where the function meets the axes on a graph, providing crucial points for sketching. In exponential functions like \(f(x) = 5^{-x}\), the x-intercept would be where \(f(x) = 0\). However, exponential functions that approach but never equal zero mean that our specific function has no x-intercepts.

• **X-intercepts**: For \(f(x) = 5^{-x}\), there are none because the function never truly hits zero.
• **Y-intercepts**: These occur when \(x = 0\). Substituting in, \(f(0) = 1\), indicating the y-intercept is at point \((0, 1)\).

Intercepts are important starting points that define where to begin your graph. Pay special attention to these points, as they give initial guidance for your sketch.

Decreasing Functions

A decreasing function is one where, as the input \(x\) increases, the output \(f(x)\) decreases. For the exponential function \(f(x) = 5^{-x}\), the negative exponent indicates that as \(x\) increases, \(-x\) also increases in negativity, shifting \(f(x)\) closer to zero.

• The base \(5\) is greater than 1, and the negative exponent flips the growth to decay.
• This decay means \(f(x)\) starts larger (closer to the y-intercept at 1) and shrinks as \(x\) grows.
• Such a property defines the graph's downward slope from left to right, visually depicted as a decreasing curve.

Understanding decreasing functions is key in correcting any errors in graphing. It emphasizes the overall trend that needs to be reflected visually.