Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$f(x)=-3^{x}+9$$

Step-by-Step Solution

Answer

Reflect \( y = 3^x \) across the x-axis and shift it up by 9 units.

Step 1: Identify the Parent Function

The given function \( f(x) = -3^x + 9 \) is derived from the parent function \( g(x) = 3^x \). The effect of the multiplication by \(-1\) is a reflection across the x-axis, and the addition of \(9\) is a vertical translation upward by 9 units.

Step 2: Analyze Transformations

The function \( f(x) = -3^x + 9 \) can be seen as the exponential function \( 3^x \) first reflected across the x-axis to become \( -3^x \), and then shifted 9 units upwards. This means each point on \( g(x) = 3^x \) is turned upside down and lifted by 9 units.

Step 3: Determine Asymptote

For \( g(x) = 3^x \), the asymptote originally is the x-axis (or \( y = 0 \)). For \( f(x) = -3^x + 9 \), the entire graph is shifted up by 9 units, meaning the new horizontal asymptote is \( y = 9 \).

Step 4: Find Key Points

To sketch the graph accurately, find a few key points. For example, when \( x = 0 \), \( f(0) = -3^0 + 9 = 8 \). When \( x = 1 \), \( f(1) = -3^1 + 9 = 6 \). When \( x = -1 \), \( f(-1) = -3^{-1} + 9 = \frac{-1}{3} + 9= 8\frac{2}{3} \). These points can be plotted to visualize the graph.

Step 5: Sketch the Graph

With the key points and asymptote identified, sketch the curve starting above the horizontal asymptote (\( y=9 \)) and showing the natural decay as \( x \) increases. Notably, the curve approaches the asymptote \( y=9 \) from below as \( x \) moves to the right.

Key Concepts

Transformation

In the world of mathematics, transformations help us to modify and shift functions in distinct ways, ensuring that we can explore a variety of function behaviors. For the function \( f(x) = -3^x + 9 \), two primary transformations occur: reflection and vertical translation.

• **Vertical Translation**: Adding or subtracting a number outside a function shifts it up or down. In \( f(x) = -3^x + 9 \), the "+9" moves the entire graph of the parent function \( 3^x \) upward by 9 units. This means every point on \( 3^x \) is moved vertically.
• **Reflection**: Reflecting a function across an axis involves flipping it over that axis. The term "\(-3^x\)" in \( f(x) = -3^x + 9 \) reflects the graph of \( 3^x \) over the x-axis, effectively turning it upside down.

These transformations allow us to understand how functions can be altered and are essential in graph analysis.

Reflection

Reflection is an interesting transformation that flips a graph across a certain line, usually an axis. In our exercise, reflection across the x-axis dramatically changes the appearance of the graph. Here's how:

• The reflection involves using the negative sign in the base of the exponent. Thus, \( f(x) = -3^x \) indicates an x-axis reflection of the parent function \( g(x) = 3^x \).
• This reflection means that all points above the x-axis in \( 3^x \) will now appear below the x-axis in \( f(x) \), and vice versa.

This phenomenon is critical for understanding changes in the overall graph orientation.

Asymptote

An asymptote is a line that a graph approaches but never actually touches. In exponential functions, identifying this line is crucial, as it helps define boundary behavior.

• For the parent function \( g(x) = 3^x \), the asymptote is the x-axis, or \( y = 0 \).
• For \( f(x) = -3^x + 9 \), after the vertical translation coupled with reflection, the horizontal asymptote changes to \( y = 9 \).

Understanding asymptotes allows us to predict where the graph will get exceedingly close but never intersect, offering insight into its tail-end behavior.

Graph Sketching

Graph sketching involves plotting a function based on key points, transformations, and asymptotic behavior. Here's a simple method to sketch the function \( f(x) = -3^x + 9 \):

• First, identify the horizontal asymptote, which is \( y = 9 \). This line acts as a guideline for where the graph will approach.
• Secondly, plot crucial points. For instance, at \( x = 0 \), the point is \( (0, 8) \); at \( x = 1 \), it's \( (1, 6) \); and at \( x = -1 \), it's approximately \( (-1, 8.67) \).
• Next, sketch the curve starting to the left of the asymptote, considering the natural decay pattern of an exponential function as \( x \) grows larger.
• Ensure the reflected nature by showing the graph approach the asymptote \( y = 9 \) from below as \( x \) increases, reflecting the inverse behavior seen in \( 3^x \).

This careful approach provides a comprehensive understanding of the overall graph shape and behavior.