Problem 66 - Beginning and Intermediate Algebra

Question

Graph the following piecewise functions. $$h(x)=\left\\{\begin{array}{cc}-\frac{2}{3} x-\frac{7}{3}, & x \geq-1 \\\2, & x<-1\end{array}\right.$$

Step-by-Step Solution

Answer

To graph the given piecewise function: 1. Identify the conditions of each piece: \(h(x) = -\frac{2}{3}x - \frac{7}{3}\) when \(x \geq -1\), and \(h(x) = 2\) when \(x < -1\). 2. Graph the first piece: Plot points for \(x \geq -1\) and connect them with a straight line. 3. Graph the second piece: Plot points for \(x < -1\) with \(h(x) = 2\) and connect them with a horizontal line. 4. Draw the final graph: Combine both graphs of the piecewise function, making sure the line for the first piece stops at \(x = -1\) and the graph of the second piece begins at the point \((-1, 2)\) and continues to the left.

Step 1: Identify the conditions for each piece of the function

The given piecewise function has two pieces. We are given the conditions for which each piece is valid: 1. \(h(x) = -\frac{2}{3}x - \frac{7}{3}\) when \(x \geq -1\) 2. \(h(x) = 2\) when \(x < -1\)

Step 2: Graph the first piece

Now, we'll graph the first piece. The function \(h(x) = -\frac{2}{3}x - \frac{7}{3}\) is a linear function with a slope of \(-\frac{2}{3}\) and a y-intercept of \(-\frac{7}{3}\). Let's use a table of values for the function when \(x \geq -1\). \[ \begin{array}{c|c} x & h(x)\\ \hline -1 & \frac{1}{3}\\ 0 & -\frac{7}{3}\\ 1 & -3 \end{array} \] We can graph these points, and then draw a line connecting the points for \(x \geq -1\).

Step 3: Graph the second piece

Now, we'll graph the second piece of the function, \(h(x) = 2\). Since \(h(x)\) is equal to a constant value of 2 when \(x < -1\), we'll be connecting all the points at which \(h(x) = 2\) in the domain \(x < -1\). Let's use a table of values: \[ \begin{array}{c|c} x & h(x) \\ \hline -2 & 2\\ -3 & 2 \end{array} \] Now we can graph these points for \(x < -1\) and draw a horizontal line at \(h(x) = 2\).

Step 4: Draw the final graph

We can now combine both graphs of the piecewise function. Include both pieces of the function in the final graph. 1. Connect the points of the first piece (\(-\frac{2}{3}x - \frac{7}{3}\)) with a straight line for all \(x \geq -1\). 2. Draw a horizontal line at \(h(x) = 2\) for all \(x < -1\). Make sure that the line for the first piece (\(x \geq -1\)) stops at \(x = -1\) and does not continue to the left. The graph of the second piece (\(h(x) = 2\) for \(x < -1\)) should begin at the point \((-1, 2)\) and continue to the left. This completes the graph of the piecewise function.

Key Concepts

Graphing Piecewise Functions

Piecewise functions are mathematical expressions defined by different equations over different parts of their domain. To graph a piecewise function, you need to understand which equation applies to each part of the domain. This involves:

• Identifying the "pieces" of the function, each corresponding to different intervals of the variable.
• Graphing each piece separately over its respective domain.
• Ensuring a clear distinction in the graph for where one piece ends and another begins, often indicated by filled or open dots.

In the provided exercise, the piecewise function has two parts:- A linear section for \(x \geq -1\)- A constant section for \(x < -1\)
By carefully plotting these sections according to their specific conditions, you will create a complete representation of the piecewise function's behavior across its domain.

Linear Functions

A linear function is a function whose graph is a straight line. It is defined by the general form \(y = mx + b\), where:

• \(m\) is the slope of the line, indicating its steepness and direction (upward or downward).
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

Linear functions are straightforward to graph because they create a straight line based on their slope and intercept.
In the given function, the linear portion is \(h(x) = -\frac{2}{3}x - \frac{7}{3}\) for \(x \geq -1\). By plotting points and using the slope, you can extend this line across its valid interval. The consistent pattern of increase or decrease is due to the function's linearity.

Function Conditions

Function conditions in piecewise functions specify the intervals where each part of the function applies. Understanding these conditions ensures that each "piece" is graphed correctly.

• Determine the domains for which each rule of the piecewise function is valid. In simpler terms, figure out when to use which equation.
• Look for inclusivity or exclusivity indicators, such as \(\leq\) or \(<\), to know whether to use a solid or open dot at boundaries.

The original problem highlights two conditions: one for \(x \geq -1\) and another for \(x < -1\). These conditions guide us in choosing which form of the function to apply and where each form begins and ends on the graph.
This clarity makes it easier for students to ensure accuracy when sketching the piecewise function on a graph.

Slope and Intercept

The slope and intercept are key components in understanding and graphing linear functions. The slope tells us how "steep" the line is and the direction it travels, while the y-intercept is where the line crosses the y-axis.

• The slope is calculated as rise over run, or the change in y divided by the change in x. A negative slope signifies a downward slope to the right, while a positive slope means the line goes upward.
• The intercept is the y-value when x is zero, marking where on the vertical axis the line begins.

The first piece of the piecewise function has a slope of \(-\frac{2}{3}\), which indicates a decline from left to right. Its y-intercept is \(-\frac{7}{3}\), which is the starting point on the axis. The combination of these parameters helps draw the line accurately for \(x \geq -1\), facilitating a clear and precise diagram of the linear portion of the piecewise function.