On Tuesday, Mr. Varner's math class filed into the room, and gawked at the message on the whiteboard: "The first student to add together all of the numbers between 1 and 100 wins four free movie tickets to the theater next Friday!" Everyone grabbed a pencil and started adding: 1 + 2 + 3 + 4 + 5... No one was further than about 20 when Brian walked in, late as usual, looked at the white board for about 15 secs, and wrote: "5050" on the bottom. The surprised Mr. Varner handed Brian the tickets and told him to take his seat.

How did he come up with the answer so fast?

### Linear and Absolute Value Function Families

In this Concept we will examine several families of functions. A **family** of functions is a set of functions whose equations have a similar form. The **parent** of the family is the equation in the family with the simplest form. For example, *y* = *x*^{2} is a parent to other functions, such as *y* = 2*x*^{2} - 5*x* + 3.

#### Linear Function Family

An equation is a member of the linear function family if it contains no powers of \begin{align*}x\end{align*} greater than 1. For example, \begin{align*}y = 2x\end{align*} and \begin{align*}y = 2\end{align*} are linear equations, while \begin{align*}y = x^2\end{align*} and \begin{align*}y = \frac{1}{x}\end{align*} are non-linear.

Linear equations are called linear because their graphs form straight lines. As you may recall from your earlier studies of algebra, we can describe any line by its average rate of change, or slope, and its *y*-intercept. (In fact, it is the constant slope of a line that makes it a line!) These aspects of a line are easiest to identify if the equation of the line is written in slope-intercept form, or *y* = *mx* + *b*. The slope of the line is equal to the coefficient *m*, and the *y*-intercept of the line is the point (0, b).

Note that a line can be a member of a family such as the family of "linear functions", and also a member of a sub-family of linear functions with the same slope. The graphs of this subfamily will be a set of parallel lines. One particular subfamily of linear functions is the **constant function** subfamily. The line *x = 5* is a constant function, as the function values are constant, or unchanging. The constant functions “sub-family” of linear functions is composed of functions whose graphs are horizontal lines.

#### Absolute Value Function Family

Let’s first consider the parent of the family: *y* = |*x*|. Because the **absolute value** of a number is that number’s distance from zero, all of the function values of an absolute value function will be non-negative. If *x* = 0, then *y* = |0| = 0. If x is positive, then the function value is equal to x. For example, the graph contains the points (1, 1), (2, 2), (3, 3), etc. However, when *x* is negative, the function value will be the opposite of the number. For example, the graph contains the points (-1, 1), (-2, 2), (-3, 3), etc. As you can see in the graph below, the absolute value function forms a “V” shape.

There are two important things to note about the graph of this kind of function. First, the absolute value graph has a vertex (a highest or lowest point) and a line of symmetry (a line that splits the function into equal and opposite 'halves'). For example, the graph of *y* = |*x*| has its vertex at (0, 0) and it is symmetric across the *y*-axis. Second, note that the graph is not curved, but composed of two straight portions. Every absolute value graph will take this shape, as long as the expression inside the absolute value is linear.

#### Piece-Wise Defined Functions

Consider again the function *y* = |*x*|. For positive *x* values, the graph resembles the identity function *y* = *x*. For negative *x* values, the graph resembles the function *y* = -*x*. We can express this relationship by defining the absolute value function in two pieces:

\begin{align*}f(x) = \begin{cases} -x, x < 0\\ x, x \ge 0\\ \end{cases}\end{align*}

We can read this notation as: the function values are equal to -*x* if *x* is negative. The function values are equal to *x* if *x* is 0 or positive.

A piece-wise defined function does not have to represent a function that can already be written as a single equation, such as the absolute value function. For example, one “piece” may be from one function family, while another piece is from a different function family.

### Examples

#### Example 1

Earlier, you were given a problem about Mr. Varner's math class giveaway.

He wrote the following message on the whiteboard: "The first student to add together all of the numbers between 1 and 100 wins four free movie tickets to the theater next Friday!" Everyone grabbed a pencil and started adding: 1 + 2 + 3 + 4 + 5... No one was further than about 20 when Brian walked in, looked at the white board for about 15 secs, and wrote: "5050" on the bottom. Mr. Varner handed Brian the tickets and told him to take his seat. How did he come up with the answer so fast?

Brian recognized that he didn't need to add each of the numbers individually, only the pairs: 100 + 0 = 100, so does 99 + 1, and 98 + 2, and so on. Since there are 50 pairs of 100, that adds up to 5000. The only number without a pair is 50, so it gets added to the total: 5050.

Function families represent this same sort of time-saver. By recognizing common bits of information and combining them in different ways, we can 'automate' some very complex-seeming processes.

#### Example 2

Identify the slope and the *y*-intercept of each line.

- \begin{align*}y=\frac{2}{3}x-1\end{align*}

This line has slope (2/3) and the *y*-intercept is the point (0, -1).

- \begin{align*}y=5\end{align*}

This is a horizontal line. The slope is 0, and the *y*-intercept is (0,5).

- \begin{align*}x=-2\end{align*}

This is a vertical line. The slope is undefined, and the line does not cross the *y*-axis. (Note that this line is not a function!)

- \begin{align*}y=\frac{2}{3}x+3\end{align*}

The slope of this line is 2/3, and the *y*-intercept is the point (0, 3).

#### Example 3

Graph the following: \begin{align*}y=|2x-1|\end{align*} and \begin{align*}y=|2x^2 -1|\end{align*}.

The graph of \begin{align*}y=|2x-1|\end{align*} makes a “V” shape, much like \begin{align*}y=|x|\end{align*}.

The function inside the absolute value, 2x+1, is linear, so the graph is composed of straight lines.

The graph of \begin{align*}y=|2x^2-1|\end{align*} is curved, and it does not have a single vertex, but two “cusps.”

The function inside the absolute value is NOT linear, therefore the graph contains curves.

#### Example 4

Sketch a graph of the function

\begin{align*}f(x) = \begin{cases} x^2, x < 2\\ x + 3, x \ge 2\\ \end{cases}\end{align*}

It is important to note that the pieces of a piece-wise defined function may or may not meet up. For example, in the graph of *f*(*x*) above, the function value is 4 at *x* = -2, but the piece of the graph that is defined by *x* + 3 is headed to the y value of 1. Therefore the two pieces do not meet.

#### Example 5

Find the x-intercepts of the function \begin{align*}f(x) = 8|x - 7| - 64\end{align*}.

To find the *x*-intercepts, set *f(x)* equal to *0*, and solve for *x*:

\begin{align*}0 = 8|x - 7| - 64\end{align*}

\begin{align*}64 = 8|x - 7|\end{align*}

\begin{align*}8 = |x - 7|\end{align*}

\begin{align*}8 = (x-7)\end{align*} *or* \begin{align*}8 = -(x-7)\end{align*}

\begin{align*}15 = x\end{align*} or \begin{align*}-1 = x\end{align*}

\begin{align*}\therefore\end{align*} the *x*-intercepts are *15 and -1*

#### Example 6

What is the graph of \begin{align*}y = |x|\end{align*}? How is that graph related to the graph of \begin{align*}y = a|x-h| + k\end{align*}? What happens to the graph of \begin{align*}y = |x|\end{align*} when the equation changes to \begin{align*}y = |x| - 5\end{align*}?

The graph of \begin{align*}y = |x|\end{align*} is shown below. You can either use a graphing tool, or plot points, noting that every positive *x* has a matching *y*, and every negative *x* matches with its positive equivalent as *y*. \begin{align*}y = |x|\end{align*} is the simplest example of a graph in the **absolute value function family**, of which \begin{align*}y = a|x-h| + k\end{align*} is the parent. Changes to \begin{align*}a\end{align*}, \begin{align*}h\end{align*}, and \begin{align*}k\end{align*} shift the graph of \begin{align*}y = |x|\end{align*} in different ways.

The graph of \begin{align*}y = |x| - 5\end{align*} is below. It is clear that the *-5* after the absolute value causes the graph to shift down 5 places.

### Review

For questions 1-5, identify the family that each function belongs to.

- \begin{align*}y = |x - 7|\end{align*}
- \begin{align*}y = 3x - 4\end{align*}
- \begin{align*}f(x) = |x^2|\end{align*}
- \begin{align*}|x| - 2 = y\end{align*}
- \begin{align*}f(x) = x + \frac{3x}{2}\end{align*}

- Graph the following piecewise function by hand: \begin{align*}f(x) = \begin{cases} x, x\geq 0\\ -x, x < 0\\ \end{cases}\end{align*}
- On your graphing calculator, graph the function \begin{align*} f(x) = |x| -2\end{align*}, and answer the following questions: a. What is the shape of the graph? b. Compare the graph to the graph in the problem above. What is the difference between the two graphs? c. What is the slope of the two lines that create the graph?

For each equation that follows, identify the coordinates of the vertex of the graph, without actually graphing.

- \begin{align*}f(x) = |6x|\end{align*}
- \begin{align*}f(x) = |x - 6| + 8\end{align*}
- \begin{align*}f(x) = |x +7| - 8\end{align*}
- \begin{align*}f(x) = |x + 5|\end{align*}

- The graph of \begin{align*}p(x) = |x|\end{align*} is shown below. If \begin{align*}t(x) = -|x|-3\end{align*}, how will the graph of \begin{align*}t(x)\end{align*} be different from the graph of \begin{align*}p(x)\end{align*}?
- Graph the absolute value equation, create your own table to justify values: \begin{align*}f(x) = |x - 3|\end{align*}
- Graph the absolute value equation, create your own table to justify values: \begin{align*}g(x) = |x+3|\end{align*}

Identify the parent function for each set of linear functions. Graph each set of functions using a graphing calculator. Identify similarities and differences of each set.

- a. \begin{align*}f(x) = x -7\end{align*}, b. \begin{align*}f(x) = x - 2\end{align*}, c. \begin{align*}f(x) = x + 1\end{align*}, d. \begin{align*}f(x) = x + 5\end{align*},

e. \begin{align*}f(x) = x + 10\end{align*}

Parent Function:

Similarities:

Differences:

- a. \begin{align*}f(x) = \frac{2}{11}x\end{align*}, b. \begin{align*}f(x) = \frac{1}{2}x\end{align*}, c. \begin{align*}f(x) = \frac{2}{3}x\end{align*}

Parent Function:

Similarities:

Differences:

- a. \begin{align*}f(x) = x -7\end{align*}, b. \begin{align*}f(x) = 2x\end{align*}, c. \begin{align*}f(x) = 4x\end{align*}, d. \begin{align*}f(x) = 2x + 5\end{align*}, e. \begin{align*}f(x) = 6x - 10\end{align*}

Parent Function:

Similarities:

Differences:

Use the standard form of a linear equation: \begin{align*}f(x) = ax + b\end{align*} and your investigations above to help you answer the following questions.

- How does the a value affect the graph?
- How does the b value affect the graph?
- How are the domain values similar/different?
- How are the range values similar/different?
- Does the a and/or b value affect the domain?
- Does the a and/or b value affect the range?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.10.