Question

Tell whether the function represents exponential growth or exponential decay. Then graph the function. \(y=\left(\frac{1}{6}\right)^x\)

Short answer

The function \(y=\left(\frac{1}{6}\right)^x\) represents exponential decay. The graph is a downward sloping curve that passes through the y-intercept at \(1\) and approaches the x-axis as \(x\) increases.

Step-by-step solution

Identify Growth or Decay

In the function \(y=\left(\frac{1}{6}\right)^x\), the base of the exponent is \(\frac{1}{6}\), which is less than \(1\). A base less than \(1\) in an exponential function indicates that the function represents exponential decay, not exponential growth.

Calculate the Y-Intercept

The y-intercept of the function occurs when \(x = 0\), because the y-coordinate when \(x = 0\) is the point where the graph crosses the y-axis. Substitute \(x = 0\) into the function: \(y=\left(\frac{1}{6}\right)^0\), which simplifies to \(y=1\). Hence, the y-intercept is \(1\).

Draw the Graph

For x > 0, the function decreases towards 0, and for x < 0, the function increases but never crosses the y-axis. The graph of \(y=\left(\frac{1}{6}\right)^x\) passes through the y-intercept at \(1\) and decays towards the x-axis as \(x\) increases. It expands upwards as \(x\) decreases but never crosses the y-axis.

Key concepts

Exponential Functions

Exponential functions are mathematical expressions that describe phenomena where growth or decay happens at a rate proportional to the current value. Such functions are characterized by the variable being in the exponent position. The general form of an exponential function is
\( y = ab^x \),
where a is the initial amount, b is the base, and x is the exponent. Growth occurs when the base b is greater than 1, leading to a rapid increase as x gets larger. In contrast, decay happens when the base b is between 0 and 1, causing the function's value to decrease as x increases.

• An exponential growth example would be a population of bacteria doubling every hour.
• An exponential decay example could be the cooling of coffee over time.

Exponential functions often model real-world scenarios such as population growth, radioactive decay, and interest in finance.

Y-Intercept

The y-intercept is an important feature of the graph of an equation, representing the point where the graph crosses the y-axis. It quantifies where the function has an input (\(x\)) of zero. For any function, we calculate the y-intercept by setting \(x = 0\) and solving for the corresponding \(y\) value.
This feature is crucial because it gives us a starting point for graphing and understanding the behavior of the function. In the context of exponential functions, the y-intercept (\(a\)) is the initial value before any growth or decay has occurred. For instance, if you invest money in an account, the y-intercept would represent your initial investment before any interest has been applied. Finding the y-intercept is a key step in sketching the graph of a function, especially when there are no other distinct points to refer to.

Graphing Exponential Functions

Graphing exponential functions involves a set of systematic steps to accurately represent the function's behavior. First and foremost, identify whether the function represents growth or decay—this determines the direction the graph will take as \(x\) increases.
Secondly, calculate the y-intercept, as this serves as a starting point on the graph. After plotting the y-intercept, choose several values of \(x\) to determine how the function behaves and plot these points as well. In the case of exponential decay, the graph will approach the x-axis asymptotically, which means it gets closer and closer but never actually touches or crosses it.
For exponential growth, the graph will climb steeply the further right it goes along the x-axis. Regardless of growth or decay, the left side of the graph will stretch vertically up or down towards infinity.
Remember to label your axes and plot a smooth curve through the points you've marked to accurately represent the function. A proper graph allows you to visualize the exponential behavior over time and can be an essential tool for understanding and communicating mathematical concepts.