Problem 30 - Algebra 2

Question

Graph each function. $$ s(t)=\left(\frac{1}{10}\right)^{t} $$

Step-by-Step Solution

Answer

The graph of the function \( s(t) = (\frac{1}{10})^{t} \) is a curve showing a rapid decrease from left to right. It passes through the points (2, 0.01), (1, 0.1), (0, 1), (-1, 10) and (-2, 100)

Step 1: Observe the function

Notice that this function is of the form \( f(t) = a^{t} \) where \( a = \frac{1}{10} \). This represents an exponential decay function

Step 2: Calculate points

For a better graph representation, calculate some points for the function. For instance, you can calculate for \( t = -2,-1,0,1,2 \). When \( t = -2 \), \( s(t) = (\frac{1}{10})^{-2} = 100 \), \( t = -1 \), \( s(t) = (\frac{1}{10})^{-1} = 10 \), \( t = 0 \), \( s(t) = (\frac{1}{10})^{0} = 1 \), \( t = 1 \), \( s(t) = (\frac{1}{10})^{1} = 0.1 \) and \( t = 2 \), \( s(t) = (\frac{1}{10})^{2} = 0.01 \)

Step 3: Plot the points

Create a graph with time \( t \) on the x-axis and \( s(t) \) on the y-axis. Plot the points calculated in the previous step on the graph. For exponential functions such as this one, the graph should show a rapid decrease from left to right because of the negative exponent

Step 4: Join the points

Join the points that have been plotted to form a smooth curve. This will be the graph of the function \( s(t) = (\frac{1}{10})^{t} \)

Key Concepts

Graphing Functions

Creating a visual representation of functions through graphing is a key mathematical skill. Graphing functions helps to better understand how the function behaves across different values. In the function given, \( s(t) = \left(\frac{1}{10}\right)^{t} \), we notice it's an exponential decay. This means that as you increase \( t \), the output of the function decreases. For graphing, the x-axis represents the variable \( t \), and the y-axis the function \( s(t) \). A crucial part of this process involves determining points on the graph by calculating function values at specific \( t \). After plotting these points, they are connected with a curve to display how the function values change. The curve's shape reveals essential characteristics of the function's behavior, such as the decay rate in exponential decay functions.

Exponential Functions

Exponential functions are an important class of functions, characterized by variables in the exponent. The standard form is \( f(x) = a^x \), where \( a \) is a positive constant. In our case, the function \( s(t) = \left(\frac{1}{10}\right)^{t} \) is a typical exponential decay function since \( a = \frac{1}{10} \), which is between 0 and 1. Exponential decay implies a rapid decrease in function values as \( t \) increases. Key properties of exponential functions include:

• Rapid change: Function values can quickly increase or decrease.
• Asymptotic behavior: As \( t\) approaches infinity, the function tends towards zero in the decay scenario.
• Decaying or growing base: If \( a > 1 \), the function grows; if \( 0 < a < 1 \), it decays.

Understanding these properties helps in plotting and predicting the behavior of the function in various contexts, like population decline or radioactive decay.

Plotting Points

Plotting points is a fundamental aspect of graphing functions. To start, you need a set of specific values for your variable \( t \). For the given function \( s(t) = \left(\frac{1}{10}\right)^{t} \), you calculate points for different \( t \) values such as -2, -1, 0, 1, and 2. These calculated points were:

• When \( t = -2 \), \( s(t) = 100 \)
• When \( t = -1 \), \( s(t) = 10 \)
• When \( t = 0 \), \( s(t) = 1 \)
• When \( t = 1 \), \( s(t) = 0.1 \)
• When \( t = 2 \), \( s(t) = 0.01 \)

Plot these points onto the graph, with \( t \) values on the horizontal axis and \( s(t) \) values on the vertical axis. Each point shows the function's value at a specific time \( t \). Once all points are plotted, connect them to form a smooth curve. This curve provides a visual representation of the exponential decay, showing how function values decrease as \( t \) increases.