Problem 36 - College Algebra

Question

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(f(x)=3^{x}\) and \(g(x)=-3^{x}\)

Step-by-Step Solution

Answer

The graph of \(f(x)=3^{x}\) is an increasing line located in the 1st and 2nd quadrants with a y-intercept at 1, whereas the graph of \(g(x)=-3^{x}\) is a decreasing line located in the 3rd and 4th quadrants with a y-intercept at -1. Neither of the graphs will touch the x-axis, they only approach it as \(x\) gets large in the negative direction.

Step 1: Plot Key Points for \(f(x)=3^{x}\)

To start, plot key points on the graph of \(f(x)=3^{x}\). These points should include the y-intercept and a few points for positive and negative x values. The y-intercept is the point where \(x=0\), and given that any number to the power of zero equals one, for \(f(x)=3^{x}\), this will be \(f(0)=3^{0}=1\). For \(x=-1\), \(x=1\), \(x=2\), we get \(1/3\), 3, 9 respectively.

Step 2: Draw \(f(x)=3^{x}\)

Based on the key points, the graph should be increasing rapidly for positive x and approaching 0 for negative x without touching the x-axis. Draw the graph by connecting these points smoothly.

Step 3: Plot Key Points for \(g(x)=-3^{x}\)

The function \(g(x)=-3^{x}\) is simply the reflection of the function \(f(x)\) across the x-axis. Therefore, the key points plotted in step 1 will have the same x-coordinates but the y-coordinates will be negated. So, for \(x=0\), the point will be at \(g(0)=-3^{0}=-1\). For \(x=-1\), \(x=1\), \(x=2\), we get \(-1/3\), -3, -9 respectively.

Step 4: Draw \(g(x)=-3^{x}\)

Draw the graph by connecting these points smoothly.The graph should be decreasing rapidly for positive x and approaching 0 for negative x without touching the x-axis.

Step 5: Validation

If available, use a graphing utility to confirm the hand-drawn graphs. Both the graphs should not touch or cross the x-axis, \(f(x)=3^{x}\) should be in the first and second quadrants, whereas \(g(x)=-3^{x}\) should be in the third and fourth quadrants.

Key Concepts

Graphing Functions

Graphing functions can seem like a daunting task, especially with exponential functions. However, understanding a few basic concepts can make it easier.
Exponential functions, like \(f(x) = 3^x\), typically increase or decrease rapidly, depending on the base and its signs.

• An exponential function with a positive base greater than one grows rapidly as \(x\) increases.
• For \(f(x) = 3^x\), when \(x < 0\), the function approaches zero but never actually touches the x-axis.
• The curve will go through the y-intercept at \(f(0) = 1\) when \(x = 0\).

To plot these functions effectively:

• Start by calculating the function's output for various \(x\)-values.
• Be sure to identify key features like intercepts and the general direction of the graph.
• Draw the curve smoothly; remember it never touches the x-axis.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is essential for graphing any mathematical function.
It consists of two perpendicular axes:

• The horizontal axis, known as the x-axis.
• The vertical axis, known as the y-axis.

These axes form four quadrants, which help in determining the position of points:

• Quadrant I: Positive \(x\) and \(y\)
• Quadrant II: Negative \(x\) and positive \(y\)
• Quadrant III: Negative \(x\) and \(y\)
• Quadrant IV: Positive \(x\) and negative \(y\)

Coordinates are written as an ordered pair \((x, y)\), where \(x\) and \(y\) are the distances from the origin along the x and y axes, respectively.
Understanding this system is vital because it provides the framework for graphing functions, enabling us to visually analyze the relationships between variables.

Reflection Across the X-Axis

Reflection across the x-axis is a transformation that produces a mirror image of a graph over the x-axis. Essentially, this kind of reflection changes the sign of the function's output values (the \(y\)-values).
For instance, graphing \(g(x) = -3^x\) is a reflection of \(f(x) = 3^x\).
This results in:

• The points originally above the x-axis (positive \(y\)-values) now below it (negative \(y\)-values).
• For \(f(x)\) with points such as \((1, 3)\) and \((2, 9)\), \(g(x)\) will have the points \((1, -3)\) and \((2, -9)\).
• The graph of \(g(x)\) is decreasing as \(x\) increases in value.

This transformation helps in solving complex problems where symmetry can simplify calculations or offer insights into the behavior of functions.