Problem 8 - Algebra A Combined Function

Question

Graph each exponential function. $$ y=\left(\frac{1}{3}\right)^{x}+2 $$

Step-by-Step Solution

Answer

Sketch an exponential decay curve approaching \( y = 2 \) from above.

Step 1: Understand the Basic Form

We start by identifying the components of the function. The given function is \[ y = \left( \frac{1}{3} \right)^{x} + 2 \]This is an exponential function where \( a = \frac{1}{3} \) and there is a vertical shift upwards by 2 units.

Step 2: Identify Key Points

Choose several values for \( x \) to calculate corresponding \( y \) values. Common points include when \( x = 0, 1, -1 \):- For \( x = 0 \), \( y = \left( \frac{1}{3} \right)^{0} + 2 = 1 + 2 = 3 \).- For \( x = 1 \), \( y = \left( \frac{1}{3} \right)^{1} + 2 = \frac{1}{3} + 2 = \frac{7}{3} \) or approximately 2.33.- For \( x = -1 \), \( y = \left( \frac{1}{3} \right)^{-1} + 2 = 3 + 2 = 5 \).

Step 3: Determine Asymptote and Behavior

The horizontal asymptote is the line that the graph approaches but never touches or crosses. For our function, as \( x \to \infty \), \( \left( \frac{1}{3} \right)^x \to 0 \), so the horizontal asymptote is \( y = 2 \).Also, as \( x \to -\infty \), \( \left( \frac{1}{3} \right)^x \) grows very large, thus \( y \to \infty \).

Step 4: Plot the Points and Asymptote

Plot the points found in Step 2 and draw the horizontal asymptote at \( y = 2 \). The graph should show exponential decay towards the asymptote as \( x \to \infty \) and rise steeply as \( x \to -\infty \).

Step 5: Draw the Curve

Using the points and understanding from previous steps, sketch a smooth curve that passes through the plotted points and approaches the asymptote \( y = 2 \) from above.

Key Concepts

Graphing Exponential Functions

Graphing an exponential function involves translating the algebraic expression into a visual format on a coordinate plane. The exponential function we are dealing with is \[ y = \left( \frac{1}{3} \right)^{x} + 2 \]When graphing these types of functions, it's important to recognize the components: the base, in this case \( \frac{1}{3} \), and any vertical shifts, like the \(+2\) here, which moves the entire graph upwards by 2 units.

• Start by choosing a range of x-values. Useful points often include \( x = 0, 1, ext{and} -1 \).
• For \( x = 0\), \( y = 3 \).
• For \( x = 1 \), \( y = \frac{7}{3} \).
• For \( x = -1 \), \( y = 5 \).

After calculating these points, plot them on the graph. Notice how the points reflect the behavior of exponential decay as \( x \) increases, and growth in the negative direction. As you plot more points, it solidifies the gradual decline towards the asymptote.

Asymptotes in Graphs

Asymptotes are essential when graphing exponential functions. They help define the boundaries that the graph will never cross. For the function \( y = \left( \frac{1}{3} \right)^{x} + 2 \), the horizontal asymptote is at \( y = 2 \). This is due to the constant \(+2\), shifting the graph upwards.

As \( x \to \infty \), the expression \( \left( \frac{1}{3} \right)^{x} \) approaches zero. Therefore, the graph hovers closer and closer to \( y = 2 \), never actually touching it. However, as \( x \to -\infty \), it grows very large because negative exponents invert the fraction.

• The asymptote acts as a visual guideline, showing where the function stabilizes.
• It forms a boundary for the graph's behavior.
• Understanding asymptotes helps predict and model the graph's progression over different x-values.

Exponential Decay

Exponential decay describes how quantities decrease rapidly and then more slowly over time or space. In the function \( y = \left( \frac{1}{3} \right)^{x} + 2 \), the decay is evident as the base \( \frac{1}{3} \) is a fraction less than one.

• As \( x \) increases, the expression raises a small fraction to a power, causing rapid shrinkage towards zero.
• This is why the graph of \( \left( \frac{1}{3} \right)^{x} \) declines as \( x \to \infty \), emphasizing the decay.
• All exponential decay functions approach a horizontal asymptote, which, in this case, is \( y = 2 \).

Understanding exponential decay can aid in grasping real-world processes like radioactive decay, where values drop quickly and then more gradually over time.