Question

Tell whether the function represents exponential growth or exponential decay. Then graph the function. \(y=(0.75)^x\)

Short answer

The function \(y=(0.75)^x\) represents exponential decay and the graph would be a downward-sloping curve passing through the points \[(-2,1.77),(-1,1.33),(0,1),(1,0.75),(2,0.56)\].

Step-by-step solution

Determine the type of function

Analyze the base of the exponential function to determine whether it is a growth or decay function. In this case, since the base of the exponent is 0.75 which is less than 1, it represents exponential decay.

Find some key points of the function

Choose a few values for \(x\) and calculate the corresponding \(y\) values. For this function, a typical set of \(x\) values might be \[-2, -1, 0, 1, 2\]. When substituting these into the function, we derive the associated \(y\) values which indicates at what points the function will pass from. Hence, for \(x=-2\) we have \(y=(0.75)^{-2}=1.77\), for \(x=-1\) we get \(y=(0.75)^{-1}=1.33\), for \(x=0\) we get \(y=(0.75)^0=1\), for \(x=1\) we get \(y=(0.75)^1=0.75\) and for \(x=2\) we get \(y=(0.75)^2=0.56\). Thus the equivalent points are \[(-2,1.77),(-1,1.33),(0,1),(1,0.75),(2,0.56)\].

Graph the function

Now, with these points, plot the graph by placing the points on the graph and then sketching the curve that passes through them. Since the function is of exponential decay, the graph will be a downward-sloping curve.

Key concepts

Exponential Functions

Exponential functions are mathematical expressions that involve a constant base raised to a variable exponent. The general form is written as \(y = a \, b^x\), where "\(a\)" represents the initial value and "\(b\)" is the base of the exponent. These functions are widely used in many fields, like finance for calculating compound interest or biology for modeling population growth and decay.

In an exponential function, the behavior of the graph is determined by the base \(b\). If \(b > 1\), the function represents exponential growth. However, if \(0 < b < 1\), it represents exponential decay. This is because when the base is less than one, repeatedly multiplying by that base reduces the value of the function as the exponent increases.

Exponential decay is observed in situations like radioactive decay or cooling of a hot object, where the quantity decreases over time. The exercise with \(y=(0.75)^x\) is an example of exponential decay because the base \(0.75\) is less than one.

Graphing Functions

Graphing functions is a crucial skill in understanding the behavior of equations visually. To graph exponential functions like \(y=(0.75)^x\), start by calculating several key points. Choose a range of \(x\) values, preferably those around zero, and calculate the corresponding \(y\) values.

For the function \(y=(0.75)^x\), you might select \(x = -2, -1, 0, 1,\) and \(2\) and compute the \(y\) values:

• For \(x = -2\), \(y = 1.77\)
• For \(x = -1\), \(y = 1.33\)
• For \(x = 0\), \(y = 1\)
• For \(x = 1\), \(y = 0.75\)
• For \(x = 2\), \(y = 0.56\)

With these points plotted on a graph, you can sketch a curve that connects them smoothly. The exponential decay of the function causes this curve to slope downward as \(x\) increases, visibly demonstrating the decrease over the range of \(x\).

Graphing not only provides visual insight into how quick the decline is but also helps in predicting future values and understanding the nature of change within the function.

Mathematical Modeling

Mathematical modeling uses mathematical structures and concepts to represent real-world systems. In particular, exponential functions can model scenarios where quantities change multiplicatively over time.

Exponential decay is found in various natural and scientific phenomena:

• Chemistry and Physics, where substances like isotopes degrade over time.
• Economics, to show diminishing returns or depreciation of assets.
• Environmental Science, when modeling the decay of pollutants.

In these models, the exponential function is a powerful tool for making predictions and understanding the dynamics of a system. For example, in the context of the exercise \(y=(0.75)^x\), this function might represent a quantity reducing by 25% sequentially, such as water temperature cooling down or a chemical reaction losing reactants.

The understanding of how base \(b\) controls the rate of decay is essential for accurately describing the behavior of such systems. Real-world observation and data collection help to adjust and validate these models, ensuring they reflect reality accurately.