Problem 16 - College Algebra Essentials

Question

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$h(x)=\left(\frac{1}{3}\right)^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(h(x) = (\frac{1}{3})^{x}\) is a decreasing curve that gets closer to the x-axis as \(x\) increases, but never touches it, and crosses the y-axis at \(h(x) = 1\).

Step 1: Identify crucial points for the table

Firstly, choose a range of \(x\) values to substitute into the function \(h(x)\). These values should include negative values, zero, and positive values. For instance, consider the set of \(x\) values {-3, -2, -1, 0, 1, 2, 3}.

Step 2: Calculate the corresponding y values

Next, compute the corresponding \(h(x)\) values. Substituting each \(x\) value into \(h(x)\) yields the respective \(h(x)\) value. Therefore, for: - \(x = -3\), \(h(x) = (\frac{1}{3})^{-3} = 27\)- \(x = -2\), \(h(x) = (\frac{1}{3})^{-2} = 9\)- \(x = -1\), \(h(x) = (\frac{1}{3})^{-1} = 3\)- \(x = 0\), \(h(x) = (\frac{1}{3})^{0} = 1\)- \(x = 1\), \(h(x) = (\frac{1}{3})^{1} = \frac{1}{3}\)- \(x = 2\), \(h(x) = (\frac{1}{3})^{2} = \frac{1}{9}\)- \(x = 3\), \(h(x) = (\frac{1}{3})^{3} = \frac{1}{27}\)

Step 3: Graph the function

Having all the coordinates, plot the points in a Cartesian plane. Draw a smooth curve through these points to represent the function \(h(x) = (\frac{1}{3})^{x}\).

Step 4: Verify the graph with a graphing utility

Use a graphing utility to verify the hand-drawn graph, ensuring that the graph generated by the software aligns with the one drawn by hand.

Key Concepts

Graphing Functions

Graphing functions is a process of visually representing a mathematical equation in a coordinate system. It allows us to see how a function behaves across different values, offering a visual interpretation of relationships between variables. To graph a function, we often start by preparing a table of values to get points that represent the function on the graph.

• Choose a range of values for the independent variable, often represented by \(x\).
• Calculate corresponding values for the dependent variable, represented by \(h(x)\) in exponential functions.

Once you have enough points, plot them on the graph. A smooth line or curve through your points shows the function's behavior.

Coordinate Geometry

Coordinate geometry revolves around placing geometric figures in a plane using a set of coordinates. Each point on the plane corresponds to a pair of numbers: the \(x\)-coordinate and \(y\)-coordinate. This system enables us to use algebraic techniques to solve geometric problems.

When graphing functions, coordinate geometry plays a pivotal role. It allows us to:

• Precisely place points based on their coordinates, facilitating accurate plotting.
• Use algebraic expressions to describe the geometric position of the graph.

By effectively using coordinate geometry, we can better understand and predict the shape and properties of graphs.

Cartesian Plane

The Cartesian plane, named after the mathematician René Descartes, is a two-dimensional surface where we plot points, lines, and curves. It is defined by two perpendicular axes: the horizontal \(x\)-axis and vertical \(y\)-axis, dividing the plane into four quadrants.

• Points are identified by pairs \((x, y)\), marking positions relative to the axes.
• The origin \((0, 0)\) is where the \(x\) and \(y\) axes intersect.

Using the Cartesian plane, we can plot exponential functions, like \(h(x)=\left(\frac{1}{3}\right)^{x}\), beautifully illustrating how values change across the plane.

Plotting Points

Plotting points involves placing points on a graph based on their coordinates. For instance, plotting the exponential function \(h(x)\) involves completing the following steps:
1. Calculate \(h(x)\) for various \(x\) values.2. Extract obtained \((x, h(x))\) pairs, such as \((-3, 27)\) and \((1, \frac{1}{3})\).3. Draw these points on the Cartesian plane corresponding to their respective coordinates.
Once all points are plotted, draw a continuous curve through these points. This curve provides a clear picture of the behavior and trend of your function, helping you understand changes in \(h(x)\) as \(x\) varies.