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<nav id="toc"><h2 class="link"><a href="frontmatter-1.html" data-scroll="frontmatter-1"><span class="title">Front Matter</span></a></h2>
<h2 class="link active"><a href="C_0.html" data-scroll="C_0"><span class="codenumber">0</span><span class="title">Preliminaries</span></a></h2>
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<li><a href="S_0_1_Functions.html" data-scroll="S_0_1_Functions" class="active">Lines, Slope, and Functions</a></li>
<li><a href="S_0_2_Exponentials.html" data-scroll="S_0_2_Exponentials">Exponential Functions</a></li>
<li><a href="S_0_3_Transformations.html" data-scroll="S_0_3_Transformations">Transformations, Compositions, and Inverses</a></li>
<li><a href="S_0_4_Logarithms.html" data-scroll="S_0_4_Logarithms">Logarithmic Functions</a></li>
<li><a href="S_0_5_TrigFunctions.html" data-scroll="S_0_5_TrigFunctions">Trigonometric Functions</a></li>
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</div></aside><main class="main"><div id="content" class="mathbook-content"><section class="section" id="S_0_1_Functions"><header title="Section 1.1 Lines, Slope, and Functions"><h1 class="heading hide-type" alt="Section 1.1 Lines, Slope, and Functions">
<span class="type">Section</span><span class="codenumber">0.1</span><span class="title">Lines, Slope, and Functions</span><a href="S_0_1_Functions.html" class="permalink">¶ permalink</a>
</h1></header><section class="introduction" id="introduction-1"><article class="objectives" id="objectives-1"><h5 class="heading"><span class="type">Objectives</span></h5>
<ul id="ul-1" style="list-style-type: disc;">
<li id="li-1"><p id="p-1">What is a function and what do we mean by its domain and range?</p></li>
<li id="li-2"><p id="p-2">What is the slope of a line? What are linear functions and families of linear functions?</p></li>
</ul></article><p id="p-3">We begin our study of calculus by reminding the reader of several pre-requisite topics. The study of calculus depends on a thorough understanding of these topics and it is imperative that the reader become as familiar as possible with these topics. In the present section we remind the reader about the concepts of functions, slope, and lines, but first, there are a few things that you should do to get your self ready to use this text.</p>
<article class="example-like" id="PA_0_1"><h5 class="heading">
<span class="type">Preview Activity</span><span class="codenumber">0.1.1</span>
</h5>
<p id="p-4">This is the first Preview Activity in this text. Your job for this activity is to get to know the textbook. </p>
<ol id="ol-1" style="list-style-type: lower-alpha;">
<li id="li-3"><p id="p-5">Where can you find the full textbook?</p></li>
<li id="li-4"><p id="p-6">What chapters of this text are you going to cover this semester. Have a look at your syllabus!</p></li>
<li id="li-5"><p id="p-7">What are the differences between Preview Activities, Activities, Examples, Exercises, Voting Questions, and WeBWork? Which ones should you do before class, which ones will you likely do during class, and which ones should you be doing after class?</p></li>
<li id="li-6"><p id="p-8">What materials in this text would you use to prepare for an exam and where do you find them?</p></li>
<li id="li-7"><p id="p-9">What should you bring to class every day?</p></li>
</ol></article></section><section class="subsection" id="subsection-1"><header title="Subsection 1.1.1 Functions"><h1 class="heading hide-type" alt="Subsection 1.1.1 Functions">
<span class="type">Subsection</span><span class="codenumber">0.1.1</span><span class="title">Functions</span>
</h1></header><p id="p-10">Let's start with the fundamental mathematical idea of a function.</p>
<article class="assemblage-like" id="definition-1">
<h5 class="heading">Function</h5>
<p id="p-11">A function is a mathematical rule that assigns exactly one output for every input.
</p>
</article>
<p id="p-12">It is easy to give many common examples of functions: </p>
<ul id="ul-2" style="list-style-type: disc;">
<li id="li-8"><p id="p-13">The area of a circle \(A\) is a function of the radius of the circle: \(A(r)= \pi r^2\text{.}\)</p></li>
<li id="li-9"><p id="p-14">The amount \(M\) in your savings account is a function of the rate of interest the bank pays as well as time.</p></li>
<li id="li-10"><p id="p-15">The fuel efficiency in your car is a function of many things, e.g. the speed at which you drive, the number of cylinders in your engine, the type of driving conditions, etc.</p></li>
<li id="li-11"><p id="p-16">The pressure on a diver is a function of the depth of the diver under water.</p></li>
</ul>
<article class="assemblage-like" id="definition-2">
<h5 class="heading">Domain of a Function</h5>
<p id="p-17">The domain is the set of all possible inputs for a function.
</p>
</article>
<article class="assemblage-like" id="definition-3">
<h5 class="heading">Range of a Function</h5>
<p id="p-18">The range is the set of all possible outputs for a function.
</p>
</article>
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<h5 class="heading">
<span class="type">Example</span>
<span class="codenumber">0.1.4</span>
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<p>
Find the domain and range of the functions \(f(x) = \sin(x), g(x) = \sqrt{x},\) and \(h(x) =
\frac{1}{x}\text{.}\)
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<p id="p-20">For \(f(x)=\sin(x)\) we recall that the sine function is defined for every possible value of \(x\) but the output is strictly between \(y=-1\) and \(y=1\text{.}\) Therefore, the domain for \(f(x) =
\sin(x)\) is \(-\infty \lt x \lt \infty\) and the range is \(-1 \le y \le 1\text{.}\) See the left plot in <a knowl="./knowl/f_0_1ex1.html" knowl-id="xref-f_0_1ex1" alt="Figure 1.1.5 " title="Figure 1.1.5 ">Figure 5</a>.</p>
<p id="p-21">For \(g(x) = \sqrt{x}\) we recall that the square root of a negative number results in an imaginary number. In this text we are interested in real-valued output for functions so we must omit all of the negative numbers from the domain and hence \(0 \le x \lt \infty\text{.}\) For the range we recall that the square root of a number will always be a non-negative number. As such, the range is \(0 \le y \lt \infty\text{.}\) See the middle plot in <a knowl="./knowl/f_0_1ex1.html" knowl-id="xref-f_0_1ex1" alt="Figure 1.1.5 " title="Figure 1.1.5 ">Figure 5</a>.</p>
<p id="p-22">For \(h(x) = \frac{1}{x}\) we recall that division by zero is mathematically impossible. That is the only troublesome point in the domain so \(-\infty \lt x \lt 0\) or \(0 \lt x \lt \infty\text{.}\) A moment's reflection also reveals that it is impossible to get zero out of the function \(h(x)\) but it is possible to get any other number. Hence \(-\infty \lt y \lt 0\) or \(0 \lt y \lt
\infty\text{.}\) See the right plot in <a knowl="./knowl/f_0_1ex1.html" knowl-id="xref-f_0_1ex1" alt="Figure 1.1.5 " title="Figure 1.1.5 ">Figure 5</a>.</p>
</p></span></span>
</div>
<!-- ***************************************************** -->
<!-- ***************************************************** -->
<!-- Figure for Example 1 -->
<figure class="figure-like" id="f_0_1ex1">
<img width="99%" src="images/0-1-ex1.svg" alt="">
<figcaption>
<span class="heading">Figure</span>
<span class="codenumber">0.1.5</span>
Graphs of the function \(f(x) = \sin(x)\text{,}\) \(g(x) = \sqrt{x}\text{,}\) and \(h(x) =
\frac{1}{x}\text{.}\)
</figcaption>
</figure>
<!-- ***************************************************** -->
<p id="p-23">It is also important to recall the notation for functions. When we write \(f(x) =
\sqrt{x}\) we are saying several things. First, the “\(f\)” is the name of the function that we're defining. The naming convention gives us a convenient way to refer to functions without having to explicitly state their algebraic form. Next, the “\((x)\)” is an explicit statement to the reader that the variable “\(x\)” is the independent variable for the function \(f\text{.}\) Lastly, the right-hand side of the definition tells us exactly what to do with the independent variable algebraically.</p>
<p id="p-24">When we write \(f(25)\) we are referring to the already defined function \(f\) and explicitly saying to replace the independent variable with the number \(25\text{.}\) In this instance, \(f(25)
= \sqrt{25} = 5\text{.}\) Similarly, if we write \(f(\sin(x))\) we mean to find the independent variable \(x\) in the function \(f\) and replace it with the function \(\sin(x)\text{.}\) In this case, \(f(\sin(x)) = \sqrt{\sin(x)}\text{.}\) This is simply a new function.</p>
<p id="p-25">A function does not always need to be given algebraically. The three primary representations of a function are the algebraic form, the graphical form, and the tabular form. For example, for the function \(f(x) = \sqrt{x}\) we are explicitly giving the algebraic form and the middle plot of <a knowl="./knowl/f_0_1ex1.html" knowl-id="xref-f_0_1ex1" alt="Figure 1.1.5 " title="Figure 1.1.5 ">Figure 5</a> shows the graphical form. <a knowl="./knowl/t_0_1ex1.html" knowl-id="xref-t_0_1ex1" alt="Table 1.1.6 " title="Table 1.1.6 ">Table 6</a> shows a portion of the table of values. The distinct disadvantage for a table of values on many functions is that there are infinitely many possible input values and a table can naturally only show finitely many of them.</p>
<figure class="figure-like" id="t_0_1ex1"><table>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">\(x\)</td>
<td class="l m b0 r0 l0 t0 lines">0</td>
<td class="l m b0 r0 l0 t0 lines">1</td>
<td class="l m b0 r0 l0 t0 lines">2</td>
<td class="l m b0 r0 l0 t0 lines">3</td>
<td class="l m b0 r0 l0 t0 lines">4</td>
<td class="l m b0 r0 l0 t0 lines">5</td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">\(f(x)\)</td>
<td class="l m b0 r0 l0 t0 lines">0</td>
<td class="l m b0 r0 l0 t0 lines">1</td>
<td class="l m b0 r0 l0 t0 lines">1.414</td>
<td class="l m b0 r0 l0 t0 lines">1.732</td>
<td class="l m b0 r0 l0 t0 lines">2</td>
<td class="l m b0 r0 l0 t0 lines">2.236</td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
</table>
<figcaption><span class="heading">Table</span><span class="codenumber">0.1.6</span>Tabular form of the function \(f(x) = \sqrt{x}\text{.}\)</figcaption></figure>
<!-- ****************************************** -->
<article class="example-like" id="A_0_1_1"><h5 class="heading">
<span class="type">Activity</span><span class="codenumber">0.1.2</span>
</h5>
<p>
The graph of a function \( f(x) \) is shown in the plot below.
</p>
<ol style="list-style-type: lower-alpha">
<li>
<p>
What is the domain of \( f(x) \)?
</p>
</li>
<li>
<p>
Approximate the range of \(f(x)\).
</p>
</li>
<li>
<p>
What are \(f(0)\), \(f(1)\), \(f(3)\), \(f(4)\), and \(f(5)\)?
</p>
</li>
</ol>
<figure>
<img src="images/0-1-act1.svg" width="80%">
</figure>
</article>
<div class="posterior">
<!-- <div id="hk-example-1" style="display: none;" class="tex2jax_ignore"><article class="example-like"><p id="p-19"> -->
<span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-hint-act1" id="hint-act1"><span class="hint"><span class="type">Hint</span></span></a></span><span id="hk-hint-act1" style="display: none;" class="tex2jax_ignore"><span class="hint">
<p>
<ol style="list-style-type: lower-alpha">
<li>
<p>
The domain is the collection of possible \(x\) values.
</p>
</li>
<li>
<p>
The range is the collection of possible \(y\) values.
</p>
</li>
<li>
<p>
Find the \(y\) values for each of the given \(x\) values.
</p>
</li>
</ol>
</p>
</p></span></span>
</div>
<div class="posterior">
<!-- <div id="hk-example-1" style="display: none;" class="tex2jax_ignore"><article class="example-like"><p id="p-19"> -->
<span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-solution-act1" id="solution-act1"><span class="solution"><span class="type">Solution</span></span></a></span><span id="hk-solution-act1" style="display: none;" class="tex2jax_ignore"><span class="solution">
<p>
<ol style="list-style-type: lower-alpha">
<li>
<p>
The domain of \(f(x)\) is \(0 \le x \le 4\).
</p>
</li>
<li>
<p>
The approximate range of \(f(x)\) is \(-7 \le y \le 3\). Without the function
itself we cannot be sure of the actual heights of the function at the maximum
and the minimum.
</p>
</li>
<li>
<p>
\(f(0) = 0\), \(f(1) = 3\), \(f(3) = -6\), \(f(4) = 0\), and \(f(5)\) does not exist
since \(5\) is not in the domain of \(f(x)\).
</p>
</li>
</ol>
</p>
</p></span></span>
</div>
<!-- ***************************************************** -->
<!-- ****************************************** -->
</section><section class="subsection" id="subsection-2"><header title="Subsection 1.1.2 Slope and Linear Functions"><h1 class="heading hide-type" alt="Subsection 1.1.2 Slope and Linear Functions">
<span class="type">Subsection</span><span class="codenumber">0.1.2</span><span class="title">Slope and Linear Functions</span>
</h1></header><p id="p-38">One of the basic graphical ideas of calculus is that if we zoom in close enough to a curved function it will look approximately linear. The words “zoom” and “close enough” will be made explicit later. We now review the features of linear functions so that the idea of “zoomed in linearity” can flow naturally later in the course.</p>
<p id="p-39">Every linear function is characterized by a constant rate of change; the slope. The slope of a linear function is a measure of the “steepness” of the line. We use the symbols \(\Delta x\) and \(\Delta y\) which mean respectively the “change in \(x\)” and the “change in \(y\)”.</p>
<!-- <article class="definition-like" id="definition-4"><h5 class="heading">
<span class="type">Definition</span><span class="codenumber">0.1.7</span><span class="title">slope</span>
</h5> -->
<article class="assemblage-like" id="definition-1">
<h5 class="heading">Slope</h5>
<p id="p-40">The slope, \(m\) of a (non-vertical) linear function \(f\) which passes through any two points \((x_1,y_1)\text{,}\) \((x_2,y_2)\) can be found using the formula</p>
\begin{equation*}
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} =
\frac{f(x_2)-f(x_1)}{x_2-x_1} = \frac{\text{ Rise } }{\text{ Run } }
\end{equation*}
</article>
<p id="p-41">As shown in <a knowl="./knowl/f_0_1slope.html" knowl-id="xref-f_0_1slope" alt="Figure 1.1.8 " title="Figure 1.1.8 ">Figure 8</a>, the slope of a linear function has the following characteristics: </p>
<ul id="ul-3" style="list-style-type: disc;">
<li id="li-21"><p id="p-42">if the line rises from left to right then the slope is positive,</p></li>
<li id="li-22"><p id="p-43">if the line falls from left to right then the slope is negative,</p></li>
<li id="li-23"><p id="p-44">if the line is horizontal then the slope is zero, and</p></li>
<li id="li-24"><p id="p-45">if the line is vertical then the slope is undefined.</p></li>
</ul>
<!-- ***************************************************** -->
<!-- Figure for Slope -->
<figure class="figure-like" id="f_0_1slope">
<img width="90%" src="images/0-1-fig3.svg" alt="">
<figcaption>
<span class="heading">Figure</span>
<span class="codenumber">0.1.8</span>
Characteristics of slope.
</figcaption>
</figure>
<!-- ***************************************************** -->
<p id="p-46">Depending on the information given there are several convenient forms of the equation of a line. Given the definition of the slope</p>
\begin{equation*}
m = \frac{y_2 - y_1}{x_2 - x_1}
\end{equation*}
<p>and letting \((x,y) = (x_2,y_2)\) be any arbitrary point we get the point-slope form of a linear function by observing that \(m = \frac{y - y_1}{x - x_1}\) which implies that \(y - y_1 = m(x-x_1)\text{.}\)</p>
<article class="assemblage-like" id="definition-5"><h5 class="heading">
Point-Slope Form of a Line
</h5>
<p id="p-47">If the linear function \(f\) has slope \(m\) and passes through the point \((x_1,y_1)\text{,}\) then the point-slope form of the equation of a line is given by:</p>
\begin{equation*}
y-y_1=m(x- x_1).
\end{equation*}
</article>
<p id="p-48">An alternate form of a linear function which is probably very familiar to most readers is the slope-intercept form of a line.</p>
<article class="assemblage-like" id="definition-6"><h5 class="heading">
Slope Intercept Form of a Line
</h5>
<p id="p-49">If the linear function \(f\) has slope \(m\) and \(y\)-intercept \(b\text{,}\) then the slope-intercept form of the equation of a line is given by:</p>
\begin{equation*}
y=mx + b.
\end{equation*}
</article>
<p id="p-50">In a calculus class the point-slope form is often the most useful. If you have a linear function written in the point-slope form you can always rearrange to get it into the slope-intercept form</p>
\begin{equation*}
y - y_1 = m(x-x_1) \implies y = mx - m x_1 + y_1.
\end{equation*}
<p id="p-51">Hence we see that the \(y\) intercept of a line can be given as \(b = -mx_1 + y_1\text{.}\) The symbols and geometry used in each of the above definitions are shown in <a knowl="./knowl/fig_0_1_linear_fn.html" knowl-id="xref-fig_0_1_linear_fn" alt="Figure 1.1.11 " title="Figure 1.1.11 ">Figure 11</a>.</p>
<figure class="figure-like" id="fig_0_1_linear_fn">
<img width="90%" src="images/0-1-fig4.svg" alt="">
<figcaption>
<span class="heading">Figure</span>
<span class="codenumber">0.1.11</span>
Anatomy of a linear function.
</figcaption>
</figure>
<!-- ***************************************************** -->
<article class="example-like" id="A_0_1_2"><h5 class="heading">
<span class="type">Activity</span><span class="codenumber">0.1.3</span>
</h5>
<p>
Find the equation of the line with the given information.
<ol style="list-style-type: lower-alpha">
<li>
<p>
The line goes through the points \((-2,5)\) and \((10,-1)\).
</p>
</li>
<li>
<p>
The slope of the line is \(3/5\) and it goes through the point \((2,3)\).
</p>
</li>
<li>
<p>
The \(y\)-intercept of the line is \((0,-1)\) and the slope is \(-2/3\).
</p>
</li>
</ol>
</p>
</article>
<div class="posterior">
<!-- <div id="hk-example-1" style="display: none;" class="tex2jax_ignore"><article class="example-like"><p id="p-19"> -->
<span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-hint-act2" id="hint-act2"><span class="hint"><span class="type">Hint</span></span></a></span><span id="hk-hint-act2" style="display: none;" class="tex2jax_ignore"><span class="hint">
<p>
<ol style="list-style-type: lower-alpha">
<li>
<p>
Recall that \(m = \frac{\Delta y}{\Delta x}\) and use the point slope form of
the line.
</p>
</li>
<li>
<p>
You are given a slope and a point. Which form of the line should you use?
</p>
</li>
<li>
<p>
You are given a slope and the \(y\) intercept. Which form of the line should
you use?
</p>
</li>
</ol>
</p>
</p></span></span>
</div>
<div class="posterior">
<!-- <div id="hk-example-1" style="display: none;" class="tex2jax_ignore"><article class="example-like"><p id="p-19"> -->
<span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-solution-act2" id="solution-act2"><span class="solution"><span class="type">Solution</span></span></a></span><span id="hk-solution-act2" style="display: none;" class="tex2jax_ignore"><span class="solution">
<p>
<ol style="list-style-type: lower-alpha">
<li>
<p>
The slope is \(m = \frac{\Delta y}{\Delta x} = \frac{5-(-1)}{(-2)-10} =
\frac{6}{-12} = -\frac{1}{2}\). Hence,
\begin{equation*}
y - 5 = -\frac{1}{2} \left( x-(-2) \right)
\end{equation*}
which can be rewritten as
\begin{equation*}
y = -\frac{1}{2} \left( x+2 \right) + 5.
\end{equation*}
</p>
</li>
<li>
<p>
Using the point-slope form of the line we get
\begin{equation*}
y - 3 = \frac{3}{5} \left( x-2 \right)
\end{equation*}
which can be rearranged to
\begin{equation*}
y = \frac{3}{5} \left( x-2 \right) + 3.
\end{equation*}
</p>
</li>
<li>
<p>
Using the slope-intercept form of the line we get
\begin{equation*}
y = -\frac{2}{3} x - 1.
\end{equation*}
</p>
</li>
</ol>
</p>
</p></span></span>
</div>
<!-- ***************************************************** -->
<!-- <div class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-example-2" id="example-2"> -->
<!-- ***************************************************** -->
<!-- EXAMPLE #2 -->
<article class="example-like">
<h5 class="heading">
<span class="type">Example</span>
<span class="codenumber">0.1.12</span>
</h5>
<p id="p-64">
<!-- </article></a></div>
<div id="hk-example-2" style="display: none;" class="tex2jax_ignore"><article class="example-like">
-->
Write the equation of the line going through the points \((5,7)\) and \((-3,2)\text{.}\)
</p>
</article>
<!-- </div> -->
<div class="posterior">
<!-- <div id="hk-example-1" style="display: none;" class="tex2jax_ignore"><article class="example-like"><p id="p-19"> -->
<span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-solution-2" id="solution-2"><span class="solution"><span class="type">Solution</span></span></a></span><span id="hk-solution-2" style="display: none;" class="tex2jax_ignore"><span class="solution"><p id="p-65">
<p id="p-65">First we calculate the slope</p>
\begin{equation*}
m = \frac{\Delta y}{\Delta x} = \frac{7 - 2}{5-(-3)} = \frac{5}{8}.
\end{equation*}
<p id="p-66">Since we have two points and neither is the \(y\) intercept of the linear function we choose to use the point-slope form of the line. Letting \((x_1,y_1) = (5,7)\) we see that</p>
\begin{equation*}
y - 7 = \frac{5}{8} \left( x-5 \right)
\end{equation*}
<p>is one form of the linear function. It is often conventient to solve for \(y\) giving us</p>
\begin{equation*}
y = \frac{5}{8} \left( x-5 \right) + 7.
\end{equation*}
<p id="p-67">Notice that we do not necessarily need to simplify all the way to the slope-intercept form of the line.</p>
</p></span></span>
</div>
<!-- ***************************************************** -->
</section><section class="subsection" id="subsection-3"><header title="Subsection 1.1.3 Linear Functions From Data"><h1 class="heading hide-type" alt="Subsection 1.1.3 Linear Functions From Data">
<span class="type">Subsection</span><span class="codenumber">0.1.3</span><span class="title">Linear Functions From Data</span>
</h1></header><p id="p-68">A feature of every linear function is that the slope is the same no matter where you are on the line. When given a table of data that you suspect might represent a linear function the slope manifests itself as a constant common difference between successive \(y\)-values.</p>
<!-- ***************************************************** -->
<!-- EXAMPLE #3 -->
<!-- <div class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-example-3" id="example-3"> -->
<article class="example-like">
<h5 class="heading">
<span class="type">Example</span>
<span class="codenumber">0.1.13</span>
</h5>
<!-- </article></a></div>
<div id="hk-example-3" style="display: none;" class="tex2jax_ignore"><article class="example-like"><p id="p-69">
-->
Consider the data in the table below.</p>
<center>
<table>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">\(x\)</td>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines">7</td>
<td class="c m b0 r0 l0 t0 lines">8</td>
<td class="c m b0 r0 l0 t0 lines">9</td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">\(y\)</td>
<td class="l m b0 r0 l0 t0 lines">12.2</td>
<td class="l m b0 r0 l0 t0 lines">17.5</td>
<td class="l m b0 r0 l0 t0 lines">22.8</td>
<td class="l m b0 r0 l0 t0 lines">28.1</td>
<td class="l m b0 r0 l0 t0 lines">33.4</td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
</table>
</center>
<p id="p-70">Demonstrate that this data is linear and write an equation that fits the data.
</p>
</article>
<!-- </div> -->
<div class="posterior">
<span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-solution-3" id="solution-3"><span class="solution"><span class="type">Solution</span></span></a></span><span id="hk-solution-3" style="display: none;" class="tex2jax_ignore"><span class="solution"><p id="p-71">
<p id="p-71">The common differences can be found for each successive \(y\)-values</p>
<table>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">\(x\)</td>
<td class="l m b0 r0 l0 t0 lines">5</td>
<td class="l m b0 r0 l0 t0 lines">6</td>
<td class="l m b0 r0 l0 t0 lines">7</td>
<td class="l m b0 r0 l0 t0 lines">8</td>
<td class="l m b0 r0 l0 t0 lines">9</td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">\(y\)</td>
<td class="l m b0 r0 l0 t0 lines">12.2</td>
<td class="l m b0 r0 l0 t0 lines">17.5</td>
<td class="l m b0 r0 l0 t0 lines">22.8</td>
<td class="l m b0 r0 l0 t0 lines">28.1</td>
<td class="l m b0 r0 l0 t0 lines">33.4</td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">Common Difference</td>
<td class="l m b0 r0 l0 t0 lines">\(\frac{17.5-12.2}{6-5} = 5.3\)</td>
<td class="l m b0 r0 l0 t0 lines">\(\frac{22.8-17.5}{7-6} = 5.3\)</td>
<td class="l m b0 r0 l0 t0 lines">\(\frac{28.1-22.8}{8-7} = 5.3\)</td>
<td class="l m b0 r0 l0 t0 lines">\(\frac{33.4-28.1}{9-8}=5.3\)</td>
<td class="l m b0 r0 l0 t0 lines">-</td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
</table>
<p id="p-72">The successive differences are clearly the same throughout the data set and the slope for this data set is \(m=5.3\text{.}\) Picking any convenient point, say \((5,12.2)\text{,}\) then allows us to write the equation of the line as</p>
\begin{equation*}
y - 12.2 = 5.3(x-5).
\end{equation*}
<p id="p-73">This could be simplified to point-slope form, but there is typically no need for this algebraic simplification.</p>
</span></span>
</div>
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<article class="example-like" id="A_0_1_3"><h5 class="heading">
<span class="type">Activity</span><span class="codenumber">0.1.4</span>
</h5>
<p>
An apartment manager keeps careful record of the rent that he charges as well as the
number of occupied apartments in his complex. The data that he has is shown in the table
below.
</p>
<center>
<table>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">Monthly Rent ($)</td>
<td class="l m b0 r0 l0 t0 lines">650</td>
<td class="l m b0 r0 l0 t0 lines">700</td>
<td class="l m b0 r0 l0 t0 lines">750</td>
<td class="l m b0 r0 l0 t0 lines">800</td>
<td class="l m b0 r0 l0 t0 lines">850</td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">Occupied Apartments</td>
<td class="l m b0 r0 l0 t0 lines">203</td>
<td class="l m b0 r0 l0 t0 lines">196</td>
<td class="l m b0 r0 l0 t0 lines">189</td>
<td class="l m b0 r0 l0 t0 lines">182</td>
<td class="l m b0 r0 l0 t0 lines">175</td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines"></td>
</tr>
</table>
</center>
<ol style="list-style-type: lower-alpha">
<li>
<p>
Just by doing simple arithmetic justify that the function relating the number of occupied
apartments and the rent is linear.
</p>
</li>
<li>
<p>
Find the linear function relating the number of occupied apartments to the rent.
</p>
</li>
<li>
<p>
If the rent were to be increased to <dollar />1000, how many occupied apartments would the
apartment manager expect to have?
</p>
</li>
<li>
<p>
At a <dollar />1000 monthly rent what net revenue should the apartment manager expect?
</p>
</li>
</ol>
</article>
<div class="posterior">
<!-- <div id="hk-example-1" style="display: none;" class="tex2jax_ignore"><article class="example-like"><p id="p-19"> -->
<span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-hint-act3" id="hint-act3"><span class="hint"><span class="type">Hint</span></span></a></span><span id="hk-hint-act3" style="display: none;" class="tex2jax_ignore"><span class="hint">
<p>
<ol style="list-style-type: lower-alpha">
<li>
<p>
Recall what we know about slope on a linear function.
</p>
</li>
<li>
<p>
Use the point slope form of the line. You should have found the slope in
part (a).
</p>
</li>
<li>
<p>
Use your answer to part (b).
</p>
</li>
<li>
<p>
How do you get the accumulated payments from all of the tenants?
</p>
</li>
</ol>
</p>
</p></span></span>
</div>
<div class="posterior">
<span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-solution-act3" id="solution-act3"><span class="solution"><span class="type">Solution</span></span></a></span><span id="hk-solution-act3" style="display: none;" class="tex2jax_ignore"><span class="solution"><p id="p-92">
<p>
<ol style="list-style-type: lower-alpha">
<li>
<p>
The slope is
\begin{equation*}
m = \frac{196-203}{700-650} = -\frac{7}{50}
\end{equation*}
and this slope is consistent no matter which pairs of points we choose.
</p>
</li>
<li>
<p>
Let \(A\) be the number of occupied apartments and let \(R\) be the rent. The
linear function relating the number of occupied apartments and the rent is
\begin{equation*}
A - 203 = -\frac{7}{50} \left( R - 650 \right).
\end{equation*}
Solving for \(A\) we get
\begin{equation*}
A(R) = -\frac{7}{50} \left( R - 650 \right) + 203.
\end{equation*}
This is a perfectly acceptable algebraic form for the answer, but if you
insist on simplifying then
\begin{equation*}
A(R) = -\frac{7}{50} R + 294.
\end{equation*}
</p>
</li>
<li>
<p>
\(A(1000) = -\frac{7}{50} \left( 1000 - 650 \right) + 203 = 154\) units
occupied.
</p>
</li>
<li>
<p>
The net revenue is the product of the monthly rent and the number of units
occupied at that rent. In this case the revenue is \( \$ 1000 \cdot 154 =
\$ 154,000\).
</p>
</li>
</ol>
</p>
</p></span></span>
</div>
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<!-- EXAMPLE #4 -->
<!-- <div class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-example-4" id="example-4"> -->
<article class="example-like">
<h5 class="heading">
<span class="type">Example</span>
<span class="codenumber">0.1.14</span>
</h5>
<!-- </article></a></div> -->
<!-- <div id="hk-example-4" style="display: none;" class="tex2jax_ignore"><article class="example-like"><p id="p-89">
-->
The Old Farmer's Almanac tells us that you can tell the temperature by counting the chirps of a cricket. It is a linear function \(T=f(C)\) given by \(T\) (in degrees Fahrenheit)=# of chirps in 15 seconds \(+40\text{.}\) We can approximate this with the formula</p>
\begin{equation*}
T = \frac{C}{4} + 40
\end{equation*}
<p>where \(C\) is the number of chirps/minute and \(T\) is in \(^\circ F\text{.}\) </p>
<ol id="ol-11" style="list-style-type: lower-alpha;">
<li id="li-46"><p id="p-90">If the chirp rate is 120 chirps/minute, what is the temperature?</p></li>
<li id="li-47"><p id="p-91">Suppose that crickets will not chirp if the temperature is below \(56^\circ F\text{.}\) We can also suppose that crickets will not chirp above \(136^\circ F\) since that is the highest temperature ever recorded at a weather station. With these parameters, what is the domain of this function?</p></li>
</ol></article>
<!-- </div> -->
<div class="posterior">
<span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-solution-4" id="solution-4"><span class="solution"><span class="type">Solution</span></span></a></span><span id="hk-solution-4" style="display: none;" class="tex2jax_ignore"><span class="solution"><p id="p-92">
<ol id="ol-12" style="list-style-type: lower-alpha;">
<li id="li-48">
<p id="p-92">If \(C = 120\) chirps/minute, substitute this into the function \(T(C)\) to obtain</p>
\begin{equation*}
T(120) = \frac{120}{4} + 40 = 30 + 40 = 70^\circ F.
\end{equation*}
</li>
<li id="li-49"><p id="p-93">To find the domain we need to find the appropriate values of \(C\) for the \(T(C)\) function. Solve \(56=C/4+40\) and get \(C = 64\text{.}\) Solve \(136=C/4+40\) and get \(C =
384\text{.}\) So the domain of \(T(C)\) is \(64 \le \text{ chirps/minute } \le 384\) or, in interval notation, \([64, 384]\text{.}\)</p></li>
</span></span>
</div>
<!-- ***************************************************** -->
</ol></section><section class="subsection" id="subsection-4"><header title="Subsection 1.1.4 Families of Linear Functions"><h1 class="heading hide-type" alt="Subsection 1.1.4 Families of Linear Functions">
<span class="type">Subsection</span><span class="codenumber">0.1.4</span><span class="title">Families of Linear Functions</span>
</h1></header><p id="p-94">We noted above that a linear function has the form \(f(x)=mx+b\text{,}\) where \(m\) is the slope of the line, and \(b\) is the \(y\)-intercept. Since \(m\) and \(b\) can take on various values, taken together, they represent a family of functions. For example, we could fix \(b = 2\text{,}\) and then draw the graphs of \(f(x)=mx+2\) for various values of \(m\text{;}\) for example, \(m = -1, -2, 2, 1\text{.}\) Doing so would give the functions in the family \(f(x)=mx+2\) shown in the left image of <a knowl="./knowl/fig_0_1_fam1.html" knowl-id="xref-fig_0_1_fam1" alt="Figure 1.1.16 " title="Figure 1.1.16 ">Figure 16</a>.</p>
<p id="p-95">Similarly, we could set \(m\) to be \(2\) and let \(b\) take on the values \(b=-1, 1, 4, -6\) and we would get some examples from the family of functions for \(y=f(x)=2x+b\) shown in the right image of <a knowl="./knowl/fig_0_1_fam1.html" knowl-id="xref-fig_0_1_fam1" alt="Figure 1.1.16 " title="Figure 1.1.16 ">Figure 16</a>.</p>
<p id="p-96">From the right image in <a knowl="./knowl/fig_0_1_fam1.html" knowl-id="xref-fig_0_1_fam1" alt="Figure 1.1.16 " title="Figure 1.1.16 ">Figure 16</a> it should be clear to the reader that parallel lines have the same slope. What can you say about the slopes of perpendicular lines? Here is the result that we state without proof.</p>
<article class="assemblage-like" id="thm_test">
<h5 class="heading">
Parallel and Perpendicular Lines
</h5>
<p id="p-97">If line \(\ell_1\) has slope \(m_1\) and line \(\ell_2\) has slope \(m_2\text{,}\) then </p>
<ul id="ul-4" style="list-style-type: disc;">
<li id="li-50"><p id="p-98">lines \(\ell_1\) and \(\ell_2\) are parallel if the slopes are the same: \(m_1 = m_2\text{,}\) and</p></li>
<li id="li-51"><p id="p-99">lines \(\ell_1\) and \(\ell_2\) are perpendicular if the slopes are opposite reciprocals: \(m_2 = -\frac{1}{m_1}\text{.}\)</p></li>
</ul></article>
<figure class="figure-like" id="fig_0_1_fam1">
<img width="99%" src="images/0-1-fig5.svg" alt="">
<figcaption>
<span class="heading">Figure</span>
<span class="codenumber">0.1.16</span>
Several members of the family of linear functions \(f(x) = mx+2\) (left) and the family \(f(x) = 2x+b\) (right).
</figcaption>
</figure>
<!--
<figure>
<img src="images/0-1-fig5.svg" width="95%">
</figure>
-->
<article class="example-like" id="A_0_1_4"><h5 class="heading">
<span class="type">Activity</span><span class="codenumber">0.1.5</span>
</h5>
<p>
Write the equation of the line with the given information.
<ol style="list-style-type: lower-alpha">
<li>
<p>
Write the equation of a line parallel to the line \(y=\frac{1}{2}x+3\) passing through
the point \((3,4)\).
</p>
</li>
<li>
<p>