Problem 39 - Algebra 1: Concepts and Skills

Question

Graph the exponential function. $$y=\left(\frac{2}{5}\right)^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(y=(\frac{2}{5})^{x}\) starts at the y-intercept (0,1), it decreases and moves towards but does not reach the 'x'-axis as 'x' increases.

Step 1: Identify the base of the exponential function

The base of the exponential function \(y=(\frac{2}{5})^{x}\) is \(\frac{2}{5}\). This is the number that is being raised to a power.

Step 2: Determine the y-intercept

The y-intercept of any exponential function \(y=b^{x}\) is always (0,1) regardless of the base 'b' because any number raised to the power of 0 equals 1. Therefore, for the given function, the y-intercept is (0,1).

Step 3: Identify the behavior of the graph

Since the base \(\frac{2}{5}\) is less than 1, the graph of the function will be decreasing. This means the graph will go downwards as 'x' increases.

Step 4: Plot a few key points

Choose a few values of 'x' to find corresponding 'y' values and plot the points. Since the function is decreasing, you can choose positive 'x' values. For instance, if 'x' = 1, 'y' = \(\frac{2}{5}\); if 'x' = 2, 'y' = \(\frac{4}{25}\), and so on.

Step 5: Draw the graph

Now, connect the plotted points to draw the graph. Ensure a smooth curve is achieved and it goes towards but does not reach the 'x'-axis as 'x' increases, demonstrating the exponential decay property.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions might seem tricky at first, but it's straightforward once you break it down. Start by understanding that an exponential function has the general form \[ y = b^x \]where 'b' is the base. To graph such a function...

• Identify the base: This tells us whether the function grows or decays as 'x' increases.
• Find the y-intercept: Set 'x' to 0 which always gives y = 1 for any positive base.
• Plot key points: Choose simple values for 'x' to calculate 'y' and get a picture of the function’s behavior.

When you've plotted several points, connect them with a smooth curve, showing the nature of the function. Remember that the curve will never touch zero if it demonstrates decay, nor infinitely rise if it shows growth. The graph becomes a powerful visual tool that tells a story of how 'y' changes with 'x'.

Exponential Decay

Exponential decay occurs when the base of the exponential function is between 0 and 1. For example, in the function \[ y = \left( \frac{2}{5} \right)^x \]the base is \(\frac{2}{5}\), which is less than 1. Here's what happens:

• The function decreases, or decays, as 'x' increases. Each step forward in the 'x' direction results in a smaller 'y'.
• The decay is rapid at the start but slows down, always approaching the 'x'-axis but never touching it.

This kind of function models situations where quantities decrease in proportion to their size, like radioactive decay. Understanding this concept can help you interpret numerous real-world phenomena.

Y-Intercept

The y-intercept of an exponential function is a crucial starting point. It’s the value of 'y' when 'x' equals 0. For any exponential function \[ y = b^x \]the y-intercept is always...\[ (0, 1) \]This happens because any positive number raised to the power of zero is 1.
It's a constant 'b' raised to the zero power rule that determines this intercept. The y-intercept is essential in graphing the function because it provides a fixed point from which the behavior of the function can be further analyzed.
By knowing this, you easily gain insight into the function's behavior right from the start.