Question

Tell whether the function represents exponential growth or exponential decay. Then graph the function. \(y=7^x\)

Short answer

The function \(y=7^x\) represents exponential growth. The graph of the function rises upwards as \(x\) increases, starting from a point close to the x-axis.

Step-by-step solution

Determine the nature of exponent

Examine the base of the exponential function. Here, the base is 7, which is greater than 1. Therefore, the function represents exponential growth.

Plot the points for function

Choose some values for \(x\) and compute corresponding \(y\) = \(7^x\). Here are some to start: when \(x=\)−1, \(y=7^{\)−1} = 1/7; when \(x=0\), \(y=7^0 = 1\); when \(x=1\), \(y=7^1 = 7\).

Graph the function

Plot the points obtained in the previous step on a graph. Because of the nature of exponential growth functions, the graph should rise upwards as \(x\) increases, starting from a point close to the x-axis.

Key concepts

Exponential Growth

Exponential growth occurs when the base of an exponential function is greater than 1. This means our function will increase as the input value, usually denoted as \( x \), increases. In the function \( y = 7^x \), the base is 7, which is greater than 1. Thus, this function exhibits exponential growth. In real-world terms, exponential growth is seen in scenarios like population growth, compound interest, and viral spread of information. They all start growing slowly, but their growth speed eventually becomes rapid. Understanding this concept helps in predicting long-term behaviors of such phenomena, allowing for better planning and decision-making.

Graphing Functions

Graphing functions is a fundamental skill in understanding how an equation behaves visually. To graph an exponential function such as \( y = 7^x \), you follow a few simple steps:

• Select a range of \( x \) values, both negative and positive.
• Calculate the corresponding \( y \) values using the function.
• Plot these points on a coordinate plane.

For example, when \( x = -1 \), \( y = \frac{1}{7} \). At \( x = 0 \), \( y = 1 \). Lastly, when \( x = 1 \), \( y = 7 \). Connect these points smoothly.For exponential growth, the graph will start close to the x-axis and rise steeply upwards. This allows visual learners to perceive the nature of growth and change in behaviors as \( x \) increases. Remember, the y-intercept of these graphs is always the base raised to the power of zero, which is 1.

Base of Exponential Function

The base of an exponential function significantly determines its behavior. It is the constant that is raised to the power \( x \) in the expression \( y = a^x \). When the base, denoted as \( a \), is greater than 1, it results in what we call exponential growth. Conversely, if \( a \) is between 0 and 1, the function will represent exponential decay. However, if the base is negative or equal to zero, the function does not qualify as an exponential one.In our example, \( y = 7^x \), the base is 7. It indicates the function's values will increase as \( x \) increases. The choice of base affects the steepness and the speed of the function’s increase or decrease. A higher base results in a faster and more pronounced growth.Understanding the implications of the base helps in foreseeing the nature of the entire function without performing extensive calculations.