Problem 7 - Algebra 2

Question

Graph each function. $$ y=8(5)^{x} $$

Step-by-Step Solution

Answer

The graph of the given exponential function \(y=8(5)^x\) passes through the points (0,8) and (1,40), rises from left to right, and has a horizontal asymptote at y=0.

Step 1: Identify the base and the vertical shift

The base of the exponential function is 5 and there is a multiplication factor of 8 that vertically shifts the graph upwards by a factor of 8.

Step 2: Identify the points

Consider the points when \(x=0\) and \(x=1\). For \(x=0\), \(y = 8(5^0) = 8\). For \(x=1\), \(y = 8(5^1) = 40\). These two points will aid in plotting the exponential function.

Step 3: Draw the graph

Begin by sketching the axes. Exponential functions have a horizontal asymptote of \(y = 0\). Use the points found to draw the curve of the function. The graph will pass through the points (0,8) and (1,40), and approaches but never touches y=0 to the left.

Step 4: Check your graph

Ensure that the shape of the curve is correct. It should start from the horizontal asymptote (y=0) on the left, rise through the point (0,8), continue rising and pass through the point (1,40) and continue rising steeply after that.

Key Concepts

Asymptote

In the context of graphing exponential functions, an asymptote is a line that the graph approaches but never actually touches or crosses. For most exponential functions, including the given function \(y = 8(5)^x\), there is a horizontal asymptote. This asymptote is usually the x-axis, or more specifically, \(y = 0\).

• As \(x\) becomes more negative, the value of \(y\) gets closer and closer to 0, meaning the graph flattens as it approaches the x-axis.
• The asymptote represents the lower boundary of the graph's range when dealing with increasing exponential functions.

Understanding where the asymptote is helps you accurately draw and interpret the behavior of the exponential curve on a graph. It shows the limit towards which the graph tends, providing insight into how the function behaves at extreme values of \(x\).

Graphing

Graphing an exponential function involves plotting it using specific calculated points and observing its general shape. In our case, the function is \(y = 8(5)^x\). Here's how you can graph it effectively:

• Identify axis and scale: Begin by drawing the x and y axes. Choose a proper scale so the significant points are easily visible.
• Determine key points: As seen in the solution, you calculate \(y\) values for specific \(x\) values, like \(x=0\) and \(x=1\).
• Plot and connect: Place the points, such as (0,8) and (1,40), on the graph. Then draw a smooth curve through the points, starting from near the horizontal asymptote \(y=0\), and moving upwards steeply.
• Indicate behavior: As the graph extends to the right, it rises sharply, demonstrating how exponential growth rapidly increases. To the left, it approaches, but never reaches \(y=0\).

Following these steps ensures you accurately represent the exponential function and can visualize its rapid growth and asymptotic behavior.

Base of Exponential

At the heart of any exponential function is its base. In this problem, the base is 5, as seen in \(y = 8(5)^x\). The base determines how quickly the exponential function grows or decays:

• An exponential function with a base greater than 1, like 5, indicates exponential growth. Influences how steeply the curve rises as \(x\) increases.
• Each increase in \(x\) results in multiplication of the previous \(y\) value by the base, illustrating its rapid expansion.

Understanding the base of an exponential is crucial as it defines the fundamental exponential growth characteristic of the function. It impacts how fast the output increases for positive inputs of \(x\) and how fast it approaches the asymptote for negative inputs.

Multiplication Factor

Unique to some exponential functions is a multiplication factor, or coefficient, as seen in the \(y = 8(5)^x\). This is the 8 in the function and it affects the graph by vertically stretching it:

• The multiplication factor adjusts the initial output of the function, seen at \(y\) when \(x=0\). Here it ensures the curve starts at a much higher point, at \(y=8\) instead of \(y=1\) if the factor were 1.
• This factor scales every y-value obtained from the base, essentially elongating the graph upwards without changing its fundamental shape.

Recognizing the multiplication factor's role helps in both accurately plotting the graph and understanding how it shifts the behavior of the curve vertically.