Precalculus by Richard Wright

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1-06 Graphs of Parent Functions

Imagine trying to fix a car with only one wrench. You might have problems because the nuts and bolts are different sizes. To do your best work on a car you need a lot of different tools in your toolbox. In a similar way, to do precalculus and higher mathematics, you need a toolbox full of functions. These are called tool-kit, or parent, functions.

Parent Functions

Constant function f(x) = c

• Domain is all real numbers.
• Range is the set {c} that contains this single element.
• Neither increasing or decreasing.
• Symmetric over the y-axis.

Linear function f(x) = x

• Domain is all real numbers.
• Range is all real numbers.
• Increases from (−∞, ∞).
• Symmetric about the origin.

Absolute value function f(x) = |x|

• Domain is all real numbers.
• Range is [0, ∞).
• Decreasing on (−∞, 0) and increasing on (0, ∞).
• Symmetric over the y-axis

Quadratic function f(x) = x²

• Domain is all real numbers.
• Range is only nonnegative real numbers, [0, ∞).
• Decreasing over (−∞, 0) and increasing on (0, ∞).
• Symmetric over the y-axis.

Cubic function f(x) = x³

• Domain is all real numbers.
• Range is all real numbers.
• Increasing on (−∞, ∞).
• Symmetric about the origin.

Reciprocal function \(f(x) = \frac{1}{x}\)

• Domain is all real numbers except 0, {x|x ≠ 0}.
• Range is all real numbers except 0, {y|y ≠ 0}.
• Decreasing on (−∞, 0) and (0, ∞).
• Symmetric about the origin and over the lines y = x and y = −x.

Reciprocal squared function \(f(x) = \frac{1}{x^2}\)

• Domain is all real numbers except 0, {x|x ≠ 0}.
• Range is only positive real numbers, (0, ∞).
• Increasing on (−∞, 0) and decreasing on (0, ∞).
• Symmetric over the y-axis.

Square root function \(f(x) = \sqrt{x}\)

• Domain is 0 or greater, [0, ∞).
• Range is 0 or greater, [0, ∞).
• Increasing on (0, ∞).
• No symmetry.

Cube root function \(f(x) = \sqrt[3]{x}\)

• Domain is all real numbers.
• Range is all real numbers.
• Increasing over (−∞, ∞).
• Symmetric about the origin.

Graph on a Graphing Calculator

All graphing calculators are different, but have similar commands. The following instructions are for the TI-84.

TI-83/84

1. Press the Y= button
2. Enter the equation
3. Press the GRAPH button
1. If the axis are not centered, press ZOOM, then choose zStandard.
2. If the graph is not visible, press ZOOM, then choose ZoomFit
3. The window range can also be set by pressing WINDOW
4. To copy the graph to your paper, press 2ND TABLE and plot the points on your paper

NumWorks

1. Press the home button and select Grapher
2. Select the Expressions tab at the top
3. Add or edit the function
4. Select the Graph tab at the top
5. The zoom options are at the top
1. Auto: should show most of the graph
2. Axes: lets you enter the values to set the visible window
3. Navigate: lets you use the arrows to move the graph around
4. Zoom with the + and − keys.
6. To copy the graph to your paper, select the Table tab at the top and plot the points on your paper

Example 1: Identify the Parent Function

Identify the parent function of f(x) = 3x³ + 1. Then graph it on a graphing calculator.

Solution

Because the function f(x) = 3x³ + 1 has x³, its parent function is cubic.

Example 2: Identify the Parent Function

Identify the parent function of \(f(x) = -\sqrt{x - 4}\). Then graph it on a graphing calculator.

Solution

Because the function \(f(x) = -\sqrt{x - 4}\) has \(\sqrt{x}\), its parent function is square root.

Graph Piecewise Functions

Piecewise functions were discussed and evaluated in lesson 01-04. Remember that they are made up of several different equations each with its own domain interval. They were evaluated by first deciding which domain the value of x was in and then evaluating that equation. Graphing piecewise functions is similar. Start by marking off each section of the domain on the x-axis. Then graph each equation only in its domain interval.

Example 3: Graph a Piecewise Function

Sketch a graph of the function.

$$ f(x) = \left\{\begin{align} x^2 &, \text{ if } x≤1 \\ 3 &, \text{ if } 1 < x ≤ 2 \\ x &, \text{ if } x > 2 \end{align}\right. $$

Solution

Before graphing the equations, start by marking the domains on the x-axis. Do something like draw light vertical dotted lines at x = 1 and x = 2.

Each of the component functions is from the library of parent functions, so their shapes are known. However, only draw the portion of the graph in each domain. At the edges of the domain, draw a filled dot when the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to sign; draw a open dot when the point is not included due to a less-than or greater-than sign. Figure 16 shows the individual pieces of the graph, then figure 17 shows the complete graph.

Analysis

Note that the graph does pass the vertical line test even at x = 1 and x = 2 because the points (1, 3) and (2, 2) are not part of the graph of the function, though (1, 1) and (2, 3) are.

Lesson Summary

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Graph on a Graphing Calculator

All graphing calculators are different, but have similar commands. The following instructions are for the TI-84.

TI-83/84

1. Press the Y= button
2. Enter the equation
3. Press the GRAPH button
1. If the axis are not centered, press ZOOM, then choose zStandard.
2. If the graph is not visible, press ZOOM, then choose ZoomFit
3. The window range can also be set by pressing WINDOW
4. To copy the graph to your paper, press 2ND TABLE and plot the points on your paper

NumWorks

1. Press the home button and select Grapher
2. Select the Expressions tab at the top
3. Add or edit the function
4. Select the Graph tab at the top
5. The zoom options are at the top
1. Auto: should show most of the graph
2. Axes: lets you enter the values to set the visible window
3. Navigate: lets you use the arrows to move the graph around
4. Zoom with the + and − keys.
6. To copy the graph to your paper, select the Table tab at the top and plot the points on your paper

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Graph a Piecewise Function

1. Mark the boundaries on the x-axis of the intervals for each piece of the domain.
2. For each piece of the domain, graph on the corresponding equation. Do not graph two functions over one interval because it would violate the criteria of a function.

Practice Exercises

Identify the parent function and then use a graphing utility to graph the function. Be sure to choose an appropriate viewing window.

1. \(f(x) = \frac{2}{3}x - \frac{1}{3}\)
2. g(x) = −x² − 4
3. \(h(x) = 2\sqrt{x}\)
4. \(j(x) = \frac{1}{x+1}\)
5. k(x) = −|x| + 4

Sketch a graph of the piecewise function.

6. \(f(x) = \left\{\begin{align} -3x - 2 &, \text{ if } x < 1 \\ \frac{1}{2}x - \frac{3}{2} &, \text{ if } x ≥ 1 \end{align}\right.\)
7. \(f(x) = \left\{\begin{align} \frac{1}{x} &, \text{ if } x < -1 \\ \sqrt{x+1}-1 &, \text{ if } x ≥ -1 \end{align}\right.\)
8. \(f(x) = \left\{\begin{align} -x^3 &, \text{ if } x < 0 \\ x^2 &, \text{ if } x ≥ 0 \end{align}\right.\)
9. \(f(x) = \left\{\begin{align} |x + 4| + 1 &, \text{ if } x ≤ 0 \\ x &, \text{ if } x > 0 \end{align}\right.\)

Identify the parent function.

10. 
11. 
12. 
13. 

Problem Solving

14. A secretary is paid $14 per hour for regular time and time-and-a-half for overtime. The weekly wage function is
    $$ W(h) = \left\{\begin{align} 14h &, \text{ if } 0 < h ≤ 40 \\ 21(h - 40) + 560 &, \text { if } h > 40\end{align}\right. $$
    where h is the number of hours worked in a week.
    1. Find W(30), W(40), W(50), W(60).
    2. The company decreased the regular work week to 35 hours. What is the new weekly wage function?

Mixed Review

15. (1-05) Use the graph of the function to estimate the intervals on which the function is increasing or decreasing.
    
16. (1-05) Find zeros of f(x) = x² − 4.
17. (1-05) Find the average rate of change from [x, x + h] for f(x) = 2x².
18. (1-04) Evaluate the function g(x) = 2x + 3 at the indicated values g(−1), g(2), g(a), g(a + h)
19. (1-04) Find the domain of the function using interval notation: \(h(x) = 3\sqrt{x - 2}\)
20. (1-02) Find the (a) radius and (b) equation of the circle with center (2, 3) and point on the circle (4, 5). Then (c) graph the circle.

Answers

1. Linear;
2. Quadratic;
3. Square root;
4. Reciprocal;
5. Absolute value;
6. 
7. 
8. 
9. 
10. Cubic
11. Reciprocal squared
12. Absolute Value
13. Square Root
14. 420, 560, 770, 980; \(W(h) = \left\{\begin{align} 14h &, \text{ if } 0 < h ≤ 35 \\ 21(h - 35) + 490 &, \text { if } h > 35\end{align}\right.\)
15. Increasing: [1, ∞), never decreases
16. −2, 2
17. 4x + 2h
18. 1, 7, 2a + 3, 2a + 2h + 3
19. [2, ∞)
20. \(2\sqrt{2}\); (x − 2)² + (y − 3)² = 8;