Precalculus by Richard Wright

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1-06 Graphs of Parent Functions

Mr. Wright teaches the lesson.

Summary: In this section, you will:

SDA NAD Content Standards (2018): PC.4.1

Toolbox
Toolbox. (pixabay.com/vinayr16)

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 toolkit, or parent, functions.

Parent Functions

Constant function f(x) = c

constant function
Constant function f(x) = 2.

Linear function f(x) = x

linear function
Linear function f(x) = x.

Absolute value function f(x) = |x|

absolute value function
Absolute value function f(x) = |x|

Quadratic function f(x) = x2

quadratic function
Quadratic function f(x) = x2

Cubic function f(x) = x3

cubic function
Cubic function f(x) = x3

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

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

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

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

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

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

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

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

TI-84
TI-84
  1. Press the o button
  2. Enter the equation
  3. Press the s button
    1. If the axes are not centered, press q, then choose zStandard.
    2. If the graph is not visible, press q, then choose ZoomFit
    3. The window range can also be set by pressing p
  4. To copy the graph to your paper, press y 0 and plot the points on your paper

NumWorks

NumWorks
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

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.

f(x) = 3x3 + 1 on a TI-84

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.

\(f(x) = -\sqrt{x - 4}\) + 1 on a NumWorks calculator

Identify the parent function of \(f(x) = -\frac{4}{x^2}\). Then graph it on a graphing calculator.

Answer

Reciprocal squared;

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.

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.

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.

marking the x-axis
The domains for each equation change 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.

x^2
3
x
(a) f(x) x2 if x ≤ 1; (b) f(x) = 3 if 1 < x ≤ 2 (c) f(x) = x if x > 2
piecewise function
\(f(x) = \left\{\begin{align} x^2 &, \text{ if } x≤1 \\ 3 &, \text{ if } 1 < x ≤ 2 \\ x &, \text{ if } x > 2 \end{align}\right.\)

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.

Sketch a graph of \(f(x) = \left\{\begin{align} x + 2 &, \text{ if } x < -1 \\ |x| &, \text{ if } x ≥ -1 \end{align}\right.\)

Answer

Lesson Summary

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

TI-84
TI-84
  1. Press the o button
  2. Enter the equation
  3. Press the s button
    1. If the axes are not centered, press q, then choose zStandard.
    2. If the graph is not visible, press q, then choose ZoomFit
    3. The window range can also be set by pressing p
  4. To copy the graph to your paper, press y 0 and plot the points on your paper

NumWorks

NumWorks
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


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.

Helpful videos about this lesson.

Practice Exercises

  1. Identify the parent function and then use a graphing utility to graph the function. Be sure to choose an appropriate viewing window.
  2. \(f(x) = \frac{2}{3}x - \frac{1}{3}\)
  3. g(x) = −x2 − 4
  4. \(h(x) = 2\sqrt{x}\)
  5. \(j(x) = \frac{1}{x+1}\)
  6. k(x) = −|x| + 4
  7. Sketch a graph of the piecewise function.
  8. \(f(x) = \left\{\begin{align} -3x - 2 &, \text{ if } x < 1 \\ \frac{1}{2}x - \frac{3}{2} &, \text{ if } x ≥ 1 \end{align}\right.\)
  9. \(f(x) = \left\{\begin{align} \frac{1}{x} &, \text{ if } x < -1 \\ \sqrt{x+1}-1 &, \text{ if } x ≥ -1 \end{align}\right.\)
  10. \(f(x) = \left\{\begin{align} -x^3 &, \text{ if } x < 0 \\ x^2 &, \text{ if } x ≥ 0 \end{align}\right.\)
  11. \(f(x) = \left\{\begin{align} |x + 4| + 1 &, \text{ if } x ≤ 0 \\ x &, \text{ if } x > 0 \end{align}\right.\)
  12. Identify the parent function.
  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?
  15. Mixed Review
  16. (1-05) Use the graph of the function to estimate the intervals on which the function is increasing or decreasing.
  17. (1-05) Find zeros of f(x) = x2 − 4.
  18. (1-05) Find the average rate of change from [x, x + h] for f(x) = 2x2.
  19. (1-04) Evaluate the function g(x) = 2x + 3 at the indicated values g(−1), g(2), g(a), g(a + h)
  20. (1-04) Find the domain of the function using interval notation: \(h(x) = 3\sqrt{x - 2}\)
  21. (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. Cubic
  7. Reciprocal squared
  8. Absolute Value
  9. Square Root
  10. 420, 560, 770, 980; \(W(h) = \left\{\begin{align} 14h &, \text{ if } 0 < h ≤ 35 \\ 21(h - 35) + 490 &, \text { if } h > 35\end{align}\right.\)
  11. Increasing: [1, ∞), never decreases
  12. −2, 2
  13. 4x + 2h
  14. 1, 7, 2a + 3, 2a + 2h + 3
  15. [2, ∞)
  16. \(2\sqrt{2}\); (x − 2)2 + (y − 3)2 = 8;