Question

Tell whether the function represents exponential growth or exponential decay. Then graph the function. \(y=(1.8)^x\)

Short answer

The function \(y = (1.8)^x\) represents an Exponential Growth. When we graph it, we get an upward-rising curve.

Step-by-step solution

Identify Type of Exponential Function

Look at the base of the exponential function, which is 1.8 in this case, as the function is given as \(y = (1.8)^x\). Since the base is greater than 1, it means the function represents exponential growth.

Graphing the Exponential Function

To graph the function \(y = 1.8^x\), choose some values for x and calculate the corresponding y values using the function. To make the work easy, consider x values -2, -1, 0, 1, 2. Their corresponding y values will be \((1.8)^{-2}\), \((1.8)^{-1}\), \((1.8)^{0}\), \((1.8)^{1}\) and \((1.8)^{2}\) respectively. Plot the (x, y) coordinates on your graph. You will have a curve rising upwards as x increases, reflecting exponential growth.

Key concepts

Exponential Growth

Exponential growth is a fascinating mathematical concept used to describe patterns where quantities rapidly increase over time. In an exponential function of the form \(y = a^x\), if the base \(a\) is greater than 1, the function will exhibit exponential growth. This is because increasing the exponent \(x\) will dramatically increase the output (\(y\)).

• When the base is greater than 1, each increase in \(x\) multiplies the current value of \(y\) by \(a\), leading to a swift rise.
• Real-world examples include population growth, compound interest, and viral spread of information.

Understanding exponential growth helps us predict trends and make informed decisions based on future projections. For instance, knowing a population grows exponentially allows for better planning in urban development and resource allocation.

Graphing Exponential Functions

Graphing an exponential function like \(y = (1.8)^x\) reveals the function's nature visually. Start by setting a range of \(x\) values and calculate corresponding \(y\) values, like -2, -1, 0, 1, and 2 for simplicity.

• For \(x = -2\) to \(x = 2\), calculate \((1.8)^{-2}\), \((1.8)^{-1}\), \((1.8)^{0}\), \((1.8)^{1}\), and \((1.8)^{2}\).
• The calculations result in coordinates that you can plot on a graph.
• Notice how the curve begins close to the x-axis for negative \(x\) values and rises sharply for positive \(x\) values.

The pattern that emerges is a rapidly ascending curve, characteristic of exponential growth. This method is useful for analyzing changes and trends across different domains, aiding in data visualization and interpretation.

Algebra 2

In Algebra 2, exponential functions and their behaviors form a crucial topic. Students gain a deeper understanding of function behavior through exploration and graphing exercises.

• Key areas include understanding the base of an exponential function and how it influences growth or decay.
• Students learn to identify and graph exponential growth and decay by observing coefficients and bases.
• Solve real-world problems involving exponential functions, such as computing interest rates or analyzing population models.

Algebra 2 provides students with the foundation needed to tackle higher-level math concepts with confidence. These skills are not only valuable in academics but also have practical applications in economics, science, and technology fields.