Problem 4 - Algebra 2

Question

Graph each function. $$ y=9(3)^{x} $$

Step-by-Step Solution

Answer

The function \( y = 9(3)^x \) can be graphed by plotting the points (-1,3), (0,9), (1,27), and (2,81) and sketching a curve that rises rapidly from left to right and that levels off as it moves right.

Step 1: Identify the Base and Coefficient

Identify the base, which is 3, and the coefficient, which is 9, in the given function \( y = 9(3)^x \). The base tells us the rate of growth, and the coefficient tells us the y-intercept.

Step 2: Create a Table of Values

Choose some values for \( x \), and calculate the corresponding \( y \) values. It is often helpful to choose a range of positive and negative numbers. For example, choose \( x = -1, 0, 1, 2 \). When \( x = -1 \), \( y = 9(3)^{-1} = 3 \). When \( x = 0 \), \( y = 9(3)^0 = 9 \). When \( x = 1 \), \( y = 9(3)^1 = 27 \). When \( x = 2 \), \( y = 9(3)^2 = 81 \). Thus, we have four points: (-1,3), (0,9), (1,27), (2,81).

Step 3: Plot the Points and Draw the Curve

Plot these points on a graph. Since we know this is an exponential function, we can sketch a curve that goes through these points. It should rise rapidly to the left of \( y-axis \) and slowly level off to the right of the \( y-axis \). The curve should never touch the \( x-axis \).

Key Concepts

Graphing Exponential Functions

Graphing exponential functions involves plotting points that represent the function's value at specific inputs, resulting in a curve that reflects the function's behavior. For the function \( y = 9(3)^x \), the graph is characterized by a rapid increase in \( y \) values as \( x \) increases. This type of growth is typical of exponential functions. The rule of thumb for these graphs is that they will never touch the \( x \)-axis, known as the horizontal asymptote. The graph starts close to this axis and rises steeply as it moves away.
For graphing an exponential function, it is crucial to realize the impact of the base and the coefficient. The base determines the rate of growth; in this case, a base of 3 indicates that each step to the right triples the \( y \) value. The coefficient, here 9, sets the initial value when \( x = 0 \), essentially influencing where the curve begins along the \( y \)-axis.
To accurately draw the graph, plotting several calculated points on the \( xy \)-plane and connecting them smoothly can help in visualizing the function's trend. Keep in mind that these functions typically display a J-shaped curve on the graph, known for exponential growth.

Exponential Growth

Exponential growth occurs when the rate of increase in the value of a quantity is proportional to its current value, resulting in the quantity growing increasingly quickly over time. In the function \( y = 9(3)^x \), exponential growth is evident. The base of 3 signifies that with every increase in \( x \) by 1, the output \( y \) is multiplied by 3.
This rapid increase in values is what makes exponential functions very powerful. They are used to model situations where growth accelerates continuously, such as population growth, compound interest, and certain scientific phenomena.
Understanding exponential growth involves recognizing its properties:

• It starts slowly and takes off rapidly.
• The larger the base, the faster the growth.
• The function will never decrease or level off.
• It only gets faster over time without bound (provided \( x \) remains positive).

Recognizing these patterns helps in predicting and understanding real-world applications of exponential growth.

Table of Values

Creating a table of values is a practical method to understand and plot an exponential function on a graph. It involves selecting specific \( x \) values, both positive and negative, to see how the function behaves and changes. For \( y = 9(3)^x \), a table of values helps in predicting how rapidly \( y \) increases as \( x \) is varied.
By computing values for \( x = -1, 0, 1, 2 \):

• \( x = -1 \): \( y = 9(3)^{-1} = 3 \)
• \( x = 0 \): \( y = 9(3)^{0} = 9 \)
• \( x = 1 \): \( y = 9(3)^{1} = 27 \)
• \( x = 2 \): \( y = 9(3)^{2} = 81 \)

We get the points \((-1, 3), (0, 9), (1, 27), (2, 81)\) which describe the shape of the graph. The values show a quick growth as \( x \) increases, with each \( y \) value being a multiple of the previous. A table helps in accurately sketching the exponential curve by providing precise points to plot. This understanding is vital for both creating the graph and interpreting the function's behavior visually.