Problem 101 - Algebra 2

Question

Sketch the graph of each function. Then locate the asymptote of the curve. $$ y=-27(3)^{x-1}+9 $$

Step-by-Step Solution

Answer

The curve of the function \(y=-27(3)^{x-1}+9\) will be an inverted and shifted exponential curve, and will have a horizontal asymptote at \(y=9\).

Step 1: Understanding the function

Firstly, acknowledge that the function provided, \(y=-27(3)^{x-1}+9\), is an exponential function. Here, \(a=-27\), \(b=3\), \(h=1\), and \(k=9\). This function implies the curve will flip over the x-axis because of the negative \(a\) value, move 1 unit right because of \(h=1\), and move up by 9 units because of \(k=9\).

Step 2: Plotting the function

Now, begin plotting some key points to start sketching the graph. For \(x=0\), we get \(y=0\). For \(x=1\), replacing it in the original formula yields \(y=9\). For \(x=2\), this gives \(y=-18\). Plot these three points and draw a smooth curve through them, reflecting the shape of an exponential function.

Step 3: Finding the asymptote

Given that this is an exponential function with a vertical shift up by 9, it will have a horizontal asymptote at \(y=9\). The curve will approach this line but never cross it.

Key Concepts

Understanding Asymptotes in Exponential Functions

An asymptote is a line that a graph approaches but never actually touches. For exponential functions, horizontal asymptotes are the most common. In the case of the function \( y = -27(3)^{x-1} + 9 \), the asymptote is influenced by the constant \( k \). This constant shifts the entire graph up or down.
For this function, \( k = 9 \). As a result, the horizontal asymptote is located at \( y = 9 \). No matter how far the graph stretches in the positive or negative x-direction, it will never reach or cross this line.
The asymptote acts as a boundary, shaping the graph's behavior as it extends. This concept is crucial in sketching and understanding the limitations or extents of exponential graphs.

Step-by-Step Graph Sketching of Exponential Functions

Graph sketching is the process of drawing a rough representation of the function's curve. For the function \( y = -27(3)^{x-1} + 9 \), the graph follows several steps to capture its essence.

• Identify Key Points: Before sketching, find key points like the y-intercept and other values. In this function, the points calculated were \((0, 0)\), \((1, 9)\), and \((2, -18)\).
• Draw the Curve: Use these points to draw a smooth, continuous curve. Make sure this curve reflects the behavior of exponential functions, where the rate of increase or decrease is proportional.
• Consider the Asymptote: The graph will approach but not cross the line \( y=9 \).

Sketching accurately also involves realizing how transformations, such as reflections and shifts, affect the general shape of the curve. Remember, the overriding behavior of exponential functions is dictated by their characteristic curve and the influence of transformations.

Impact of Transformations on Exponential Functions

Transformations alter the appearance of a function's graph without changing its essential shape. The function \( y = -27(3)^{x-1} + 9 \) includes several transformations that modify its graph.

• Reflection: The negative value of \( a \) flips the graph over the x-axis. This reflection changes the direction in which the exponential curve opens.
• Horizontal Shift: The term \( x-1 \) shifts the graph 1 unit to the right. In general, \( x-h \) in the function \((x-h)\) represents a shift of \( h \) units to the right.
• Vertical Shift: The constant \( +9 \) results in a shift 9 units upwards, hence changing the baseline from which the graph grows or decays.

These transformations need careful consideration in predicting the function's appearance, ensuring the resulting graph reflects these changes accurately. Understanding transformations helps in manipulating the function to fit different scenarios or situations.