section-1-4.xml

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<section xml:id="slope-and-rate-of-change"   xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Slope and Rate of Change</title>
<introduction></introduction>
<subsection><title>Using Ratios for Comparison</title>   
<p>Which is more expensive, a 64-ounce bottle of Velvolux dish soap that costs <dollar />3.52, or a 60-ounce bottle of Rainfresh dish soap that costs <dollar />3.36?</p>
<p>You are probably familiar with the notion of comparison shopping. To decide which dish soap is the better buy, we compute the unit price, or price per ounce, for each bottle. The unit price for Velvolux is
    <me>\frac{352 \text{ cents}}{64 \text{ ounces}}= 5.5 \text{ cents per ounce}</me>
and the unit price for Rainfresh is
    <me>\frac{336 \text{ cents}}{60 \text{ ounces}}= 5.6 \text{ cents per ounce}</me>
</p>
<p>
    The Velvolux costs less per ounce, so it is the better buy. By computing the price of each brand for <em>the same amount of soap</em>, it is easy to compare them.</p>
<p>
    In many situations, a ratio, similar to a unit price, can provide a basis for comparison. <xref ref="example-rate-of-growth" text="type-global" /> uses a ratio to measure a rate of growth.
</p>

<example xml:id="example-rate-of-growth"><p>Which grow faster, Hybrid A wheat seedlings, which grow 11.2 centimeters in 14 days, or Hybrid B seedlings, which grow 13.5 centimeters in 18 days?</p>
<solution><p>
    We compute the growth rate for each strain of wheat. Growth rate is expressed as a ratio, <m>\dfrac{\text{centimeters}}{\text{days}}</m>, or centimeters per day. The growth rate for Hybrid A is
    <me>\frac{11.2 \text{ centimeters}}{14 \text{ days}}= 0.8 \text{ centimeters per day}</me>
and the growth rate for Hybrid B is
    <me>\frac{13.5 \text{ centimeters}}{18 \text{ days}}= 0.75 \text{ centimeters per day}</me>
Because their rate of growth is larger, the Hybrid A seedlings grow faster.
</p></solution>
</example>

<p>
    By computing the growth of each strain of wheat seedling over the same unit of time, a single day, we have a basis for comparison. In this case, the ratio <m>\dfrac{\text{centimeters}}{\text{day}}</m> measures the rate of growth of the wheat seedlings.
</p>

<exercise xml:id="exercise-gas-mileage"><statement><p>
    Delbert traveled <m>258</m> miles on <m>12</m> gallons of gas, and Francine traveled <m>182</m> miles on <m>8</m> gallons of gas. Compute the ratio <m>\dfrac{\text{miles}}{\text{gallon}}</m> for each car. Whose car gets the better gas mileage?
</p></statement> 
<answer><p>Delbert: <m>21.5</m> mpg, Francine: <m>22.75</m> mpg. Francine gets better mileage.</p></answer>  
</exercise>

<p>
    In <xref ref="exercise-gas-mileage" text="type-global" />, the ratio <m>\frac{\text{miles}}{\text{gallon}}</m> measures the rate at which each car uses gasoline. By computing the mileage for each car for the same amount of gas, we have a basis for comparison. We can use this same idea, finding a common basis for comparison, to measure the steepness of an incline.
</p>
</subsection>

<subsection><title>Measuring Steepness</title>
<p>Imagine you are an ant carrying a heavy burden along one of the two paths shown in <xref ref="fig-ant-climb" text="type-global" />. Which path is more difficult? Most ants would agree that the steeper path is more difficult. </p>
<sidebyside><p>But what exactly is steepness? It is not merely the gain in altitude, because even a gentle incline will reach a great height eventually. Steepness measures how sharply the altitude increases. An ant finds the second path more difficult, or steeper, because it rises 5 feet while the first path rises only 2 feet over the same horizontal distance.</p>
<figure xml:id="fig-ant-climb"><caption></caption><image source="images/fig-ant-climb"   width="110%"><description>ant's climb</description></image></figure>
</sidebyside>

<p>To compare the steepness of two inclined paths, we compute the ratio of change in
altitude to change in horizontal distance for each path.</p>

<example xml:id="example-steep-trail">
    <p>Which is steeper, Stony Point trail, which climbs 400 feet over a horizontal distance of 2500 feet, or Lone Pine trail, which climbs 360 feet over a horizontal distance of 1800 feet?</p>

<solution>
    <p>For each trail, we compute the ratio of vertical gain to horizontal distance. For Stony Point trail, the ratio is
    <me>\frac{400 \text{ feet}}{2500 \text{ feet}}= 0.16 </me>  
    and for Lone Pine trail, the ratio is
    <me>\frac{360 \text{ feet}}{1800 \text{ feet}}= 0.20 </me> 
    Lone Pine trail is steeper, because it has a vertical gain of 0.20 foot for every foot traveled horizontally. Or, in more practical units, Lone Pine trail rises 20 feet for every 100 feet of horizontal distance, whereas Stony Point trail rises only 16 feet over a horizontal distance of 100 feet.</p>
</solution>
</example>

<exercise xml:id="exercise-steep-steps">
    <statement><p>Which is steeper, a staircase that rises <m>10</m> feet over a horizontal distance of <m>4</m> feet, or the steps in the football stadium, which rise <m>20</m> yards over a horizontal distance of <m>12</m> yards?</p></statement>
    <answer><p>The staircase is steeper.</p></answer>
</exercise>
</subsection>

<subsection><title>Definition of Slope</title>
<p>
    To compare the steepness of the two trails in <xref ref="example-steep-trail" text="type-global" />, it is not enough to know which trail has the greater gain in elevation overall. Instead, we compare their elevation gains over the same horizontal distance. Using the same horizontal distance provides a basis for comparison. The two trails are illustrated in <xref ref="fig-trail-climb-grid" text="type-global" /> as lines on a coordinate grid.
</p>    
<figure xml:id="fig-trail-climb-grid"><caption></caption><image source="images/fig-trail-climb-grid"   width="80%"><description>trails on grid</description></image></figure>
<p>
    The ratio we computed in <xref ref="example-steep-trail" text="type-global" />,
    <me>\frac{\text{change in elevation}}{\text{change in horizontal position}}</me>  
    appears on the graphs in <xref ref="fig-trail-climb-grid" text="type-global" /> as
    <me>\frac{\text{change in }y\text{-coordinate}}{\text{change in }x\text{-coordinate}}</me> 
    For example, as we travel along the line representing Stony Point trail, we move from the point<m>0, 0)</m> to the point <m>(2500, 400)</m>. The <m>y</m>-coordinate changes by <m>400</m> and the <m>x</m>-coordinate changes by <m>2500</m>, giving the ratio <m>0.16</m> that we found in <xref ref="example-steep-trail" text="type-global" />2. We call this ratio the <term>slope</term> of the line.
</p>

<assemblage><title>Definition of Slope</title>
    <p>The <term>slope</term><idx>slope</idx> of a line is the ratio
    <me>\frac{\text{change in }y\text{-coordinate}}{\text{change in }x\text{-coordinate}}</me> 
    as we move from one point to another on the line.
    </p>
</assemblage>

<example xml:id="example-slope-grid">
<p>
    Compute the slope of the line that passes through points <m>A</m> and <m>B</m> in <xref ref="fig-slope-grid" text="type-global" />.
</p>
<figure xml:id="fig-slope-grid"><caption></caption><image source="images/fig-slope-grid"   width="35%"><description>trails on grid</description></image></figure>

<solution><p>
    As we move along the line from <m>A</m> to <m>B</m>, the <m>y</m>-coordinate changes by <m>3</m> units, and the <m>x</m>-coordinate changes by <m>4</m> units. The slope of the line is thus
    <me>\frac{\text{change in }y\text{-coordinate}}{\text{change in }x\text{-coordinate}}=\frac{3}{4}</me> 
</p></solution>
</example>

<exercise xml:id="exercise-slope-grid"><statement>
    <sidebyside widths="50% 40%" valigns="middle middle"><p>Compute the slope of the line through the indicated points in <xref ref="fig-slope-grid2" text="type-global" />. On both axes, one square represents one unit. 
    <me>\frac{\text{change in }y\text{-coordinate}}{\text{change in }x\text{-coordinate}}=</me> 
    </p>
    <figure xml:id="fig-slope-grid2"><caption></caption><image source="images/fig-slope-grid2"><description>line on grid</description></image></figure>
    </sidebyside>
</statement>
<answer><p><m>\dfrac{1}{4} </m></p></answer>
</exercise>

<p>
    The slope of a line is a number. It tells us how much the <m>y</m>-coordinates of points on the line increase when we increase their <m>x</m>-coordinates by 1 unit. For instance, the slope <m>\frac{3}{4}</m> in <xref ref="example-slope-grid" text="type-global" /> means that the <m>y</m>-coordinate increases by <m>\frac{3}{4}</m> unit when the <m>x</m>-coordinate increases by 1 unit. For increasing graphs, a larger slope indicates a greater increase in altitude, and hence a steeper line.
</p>
</subsection>

<subsection><title>Notation for Slope</title>
<p>
    We use a shorthand notation for the ratio that defines slope,
    <me>\frac{\text{change in }y\text{-coordinate}}{\text{change in }x\text{-coordinate}}</me> 
    The symbol <m>\Delta</m> (the Greek letter delta) is used in mathematics to denote <em>change in</em>. In particular, <m>\Delta y</m> means <em>change in <m>y</m>-coordinate</em>, and <m>\Delta x</m> means <em>change in <m>x</m>-coordinate</em>. We also use the letter <m>m</m> to stand for slope. With these symbols, we can write the definition of slope as follows.
</p>

<assemblage><title>Notation for Slope</title>
<p>
    The <term>slope</term><idx>slope</idx> of a line is given by
    <me>\frac{\Delta y}{\Delta x}=\frac{\text{change in }y\text{-coordinate}}{\text{change in }x\text{-coordinate}}\text{, }~x\ne 0</me></p><p>

    <sidebyside width="35%"><image source="images/fig-slope-Delta"><description>slope of line</description></image></sidebyside></p>
</assemblage>

<example xml:id="example-Great-Pyramid">
    <p>
        The Great Pyramid of Khufu in Egypt was built around 2550 B.C. It is 147 meters tall and has a square base 229 meters on each side. Calculate the slope of the sides of the pyramid, rounded to two decimal places.
    </p>
    <solution><sidebyside>
        <p>
            From <xref ref="fig-Great-Pyramid" text="type-global" />, we see that <m>\Delta x</m> is only half the base of the Great Pyramid, so <me>\Delta x = 0.5(229) = 114.5</me>
            and the slope of the side is 
            <me>m=\frac{\Delta y}{\Delta x}= \frac{147}{114.5} = 1.28</me>
        </p>
        
        <figure xml:id="fig-Great-Pyramid"><caption></caption><image source="images/fig-Great-Pyramid"   width="90%"><description>Great Pyramid of Khufu</description></image></figure>
    </sidebyside>
    </solution>
</example>

<exercise xml:id="Kukulcan-pyramid"><statement>
    <p>The Kukulcan Pyramid at Chichen Itza in Mexico was built around 800 A.D. It is <m>24</m> meters high, with a temple built on its top platform, as shown in <xref ref="fig-Kukulcan-Pyramid" text="type-global" />.</p>
    <sidebyside widths="45% 50%"><p>
          The square base is <m>55</m> meters on each side, and the top platform is <m>19.5</m> meters on each side. Calculate the slope of the sides of the pyramid. Which pyramid is steeper, Kukulcan or the Great Pyramid?
    </p>
    <figure xml:id="fig-Kukulcan-Pyramid"><caption></caption><image source="images/fig-Kukulcan-Pyramid"   width="110%"><description>Kukulcan Pyramid of Chichen Itza</description></image></figure>
    </sidebyside>
</statement>
<answer><p><m>1.35</m>; Kukulcan is steeper.</p></answer>
</exercise>

<p>
    So far, we have only considered examples in which <m>\Delta x</m> and <m>\Delta y</m> are positive numbers, but they can also be negative.
    <me>\Delta x = \begin{cases}
        \text{positive if } x \text{ increases (move to the right)}\\
        \text{negative if } x \text{ decreases (move to the left)}\\
        \end{cases}
    </me>
    <me>\Delta y = \begin{cases}
        \text{positive if } y \text{ increases (move up)}\\
        \text{negative if } y \text{ decreases (move down)}~~~~~~~~~\\
        \end{cases}
    </me>
</p>

<example xml:id="example-negative-slope">
    <sidebyside widths="45% 50%"><p>
        Compute the slope of the line that passes through the points <m>P(-4, 2)</m> and <m>Q(5, -1)</m> shown in <xref ref="fig-negative-slope" text="type-global" /> Illustrate <m>\Delta y</m> and <m>\Delta x</m> on the graph.
    </p>
    <figure xml:id="fig-negative-slope"><caption></caption><image source="images/fig-negative-slope"   width="85%"><description>negative slope</description></image></figure>
    </sidebyside>
    <solution><p>As we move from the point <m>P(-4, 2)</m> to the point <m>Q(5, -1)</m>, we move <m>3</m> units <em>down</em>, so <m>\Delta y = -3</m>. We then move <m>9</m> units to the right, so <m>\Delta x = 9</m>.
        Thus, the slope is
        <me>m = \frac{\Delta y}{\Delta x}=\frac{-3}{9}=\frac{-1}{3}</me>
        <m>\Delta y</m> and <m>\Delta x</m> are labeled on the graph.
    </p></solution>
</example>

<p>We can move from point to point in either direction to compute the slope. The line graphed in <xref ref="example-negative-slope" text="type-global" /> decreases as we move from left to right and hence has a negative slope.</p> 
<sidebyside widths="45% 50%"><p>
    The slope is the same if we move from point <m>Q</m> to point <m>P</m> instead of from <m>P</m> to <m>Q</m>. (See <xref ref="fig-negative-slope2" text="type-global" />.) In that case, our computation looks like this:
        <me>m = \frac{\Delta y}{\Delta x}=\frac{3}{-9}=\frac{-1}{3}</me>
        <m>\Delta y</m> and <m>\Delta x</m> are labeled on the graph.
</p>
    <figure xml:id="fig-negative-slope2"><caption></caption><image source="images/fig-negative-slope2"   width="85%"><description>negative slope</description></image></figure>
    </sidebyside>   
</subsection>

<subsection><title>Lines Have Constant Slope</title>
<p>
    How do we know which two points to choose when we want to compute the slope of a line? It turns out that any two points on the line will do.
</p>    

<exercise xml:id="exercise-slope-any-points"><statement>
    <ol label="a">
        <li><p>Graph the line <m>4x - 2y = 8</m> by finding the <m>x</m>- and <m>y</m>-intercepts</p></li>
        <li><p>Compute the slope of the line using the <m>x</m>-intercept and <m>y</m>-intercept.</p></li>
        <li><p>Compute the slope of the line using the points <m>(4, 4)</m> and <m>(1, -2)</m>.</p></li>
    </ol>
</statement>
<answer><p>
    <ol label="a" cols="3">
        <li><p><image source="images/fig-in-ex-ans-1-4-5"  width="80%"><description>line</description></image> </p></li>
        <li><p><m>2</m></p></li>
        <li><p><m>2</m></p></li>
    </ol>
</p></answer>
</exercise>

<note><p>
    <xref ref="exercise-slope-any-points" text="type-global"/> illustrates an important property of lines: They have constant slope. No matter which two points we use to calculate the slope, we will always get the same result. We will see later that lines are the only graphs that have this property.</p></note>
    <p> We can think of the slope as a scale factor that tells us how many units <m>y</m> increases (or decreases) for each unit of increase in <m>x</m>. Compare the lines in <xref ref="fig-slope-scale-factor" text="type-global" />
    <figure xml:id="fig-slope-scale-factor"><caption></caption><image source="images/fig-slope-scale-factor"   width="90%"><description>slopes</description></image></figure>
    Observe that a line with positive slope increases from left to right, and one with negative slope decreases. What sort of line has slope <m>m = 0</m>?
</p>
</subsection>

<subsection><title>Meaning of Slope</title>
<sidebyside widths="50% 35%" valigns="middle middle"><p>
    In <xref ref="example-Annelise" text="phrase-hybrid"/>, we graphed the equation <m>C = 5 + 3t</m> showing the cost of a bicycle rental in terms of the length of the rental. The graph is reproduced in <xref ref="fig-meaning-slope" text="type-global" />. We can choose any two points on the line to compute its slope. Using points <m>P</m> and <m>Q</m> as shown, we find that
    <me>m = \frac{\Delta C}{\Delta t}= \frac{9}{3}=3</me>
    The slope of the line is <m>3</m>.
</p>
<figure xml:id="fig-meaning-slope"><caption></caption><image source="images/fig-meaning-slope"><description>meaning of slope</description></image></figure></sidebyside>
<p>
    What does this value mean for the cost of renting a bicycle? The expression
    <me>\frac{\Delta C}{\Delta t}= \frac{9}{3}</me>
    stands for
    <me>\frac{\text{change in cost}}{\text{change in time}}= \frac{9 \text{ dollars }}{3 \text{ hours}}</me>
</p>
<p>
    If we increase the length of the rental by 3 hours, the cost of the rental increases by 9 dollars. The slope gives the rate of increase in the rental fee, 3 dollars per hour. In general, we can make the following statement.
</p>

<assemblage><title>Rate of Change</title>
<p>
    The slope of a line measures the <em>rate of change</em> of the output variable with respect to the input variable.
</p>  
</assemblage>

<p>
    Depending on the variables involved, this rate might be interpreted as a rate of growth or a rate of speed. A negative slope might represent a rate of decrease or a rate of consumption. The slope of a graph can give us valuable information about the variables.
</p>

<example xml:id="example-truck-speed">
    <sidebyside widths="50% 30%" valigns="middle middle"><p>The graph in <xref ref="fig-truck-speed" text="type-global"/> shows the distance in miles traveled by a big-rig truck driver after <m>t</m> hours on the road. 
        <ol label="a">
            <li>Compute the slope of the graph.</li>
            <li>What does the slope tell us about the problem?</li>
        </ol>
    </p>
    <figure xml:id="fig-truck-speed"><caption></caption><image source="images/fig-truck-speed"><description>truck</description></image></figure>
</sidebyside>
    <solution><p>
        <ol label="a">
            <li>Choose any two points on the line, say <m>G(2, 100)</m> and <m>H(4, 200)</m>, in <xref ref="fig-truck-speed" text="type-global" />. As we move from <m>G</m> to <m>H</m>, we find
            <me>m = \frac{\Delta D}{\Delta t}=\frac{100}{2}=50</me>
            The slope of the line is 50.</li>
            <li>The best way to understand the slope is to include units in the calculation. For our example,
            <me>\frac{\Delta D}{\Delta t} \text{ means } \frac{\text{change in distance}}{\text{change in time}}</me>
            or
            <me>\frac{\Delta D}{\Delta t}
                =\frac{100 \text{ miles}}{2\text{ hours}}
                =50 \text{ miles per hour}</me>
                The slope represents the trucker’s average speed or velocity.</li>
        </ol>
    </p></solution>
</example>

<exercise xml:id="example-ski-lift"><statement><p>
    The graph in <xref ref="fig-ski-lift" text="type-global"/> shows the altitude, <m>a</m> (in feet), of a skier <m>t</m> minutes after getting on a ski lift.
    <ol label="a">
        <li><p>Choose two points and compute the slope (including units).</p></li>
        <li><p>What does the slope tell us about the problem?</p></li>
    </ol>
    <figure xml:id="fig-ski-lift"><caption></caption><image source="images/fig-ski-lift"   width="40%"><description>ski lift graph</description></image></figure>
</p></statement>
<answer><p>
    <ol label="a" cols="2">
        <li><p><m>150</m></p></li>
        <li><p>Altitude increases by <m>150</m> feet per minute.</p></li>
    </ol>
</p></answer>
</exercise>
</subsection>

<subsection><title>A Formula for Slope</title>
<p>
    We have defined the slope of a line to be the ratio <m>m = \frac{\Delta y}{\Delta x}</m> as we move from one point to another on the line. So far, we have computed <m>\Delta y</m> and <m>\Delta x</m> by counting squares on the graph, but this method is not always practical. All we really need are the coordinates of two points on the graph.
</p>
<p>We will use <term>subscripts</term><idx>subscripts</idx> to distinguish the two points:</p>
<p><m>P_1</m> means <em>first point</em> and <m>P_2</m> means <em>second point</em>.</p>
<p>We denote the coordinates of <m>P_1</m> by <m>(x_1, y_1)</m> and the coordinates of <m>P_2</m> by <m>(x_2, y_2)</m>.
</p>

<sidebyside widths="40% 50%">
   <figure xml:id="fig-slope-formula"><caption></caption><image source="images/fig-slope-formula"><description>slope of line</description></image></figure>

    <p>
        Now consider a specific example. The line through the two points <m>P_1 (2, 9)</m> and <m>P_2 (7,-6)</m> is shown in <xref ref="fig-slope-formula" text="type-global"/>. We can find <m>\Delta x</m> by subtracting the <m>x</m>-coordinates of the points: 
        <me>\Delta x = 7 - 2 = 5 </me>
        In general, we have 
        <me>\Delta x = x_2 - x_1 </me>
        and similarly
        <me>\Delta y = y_2 - y_1 </me>
    </p>
</sidebyside>
<p>
    These formulas work even if some of the coordinates are negative; in our example 
    <me>\Delta y = y_2 - y_1 = -6 - 9 = -15</me>
    By counting squares <em>down</em> from <m>P_1</m> to <m>P_2</m>, we see that <m>\Delta y</m> is indeed <m>-15</m>. The slope of the line in <xref ref="fig-slope-formula" text="type-global" /> is </p>
    <p>\begin{align*}
            m<ampersand />=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1} 
            <backslash /><backslash />
            <ampersand />=\frac{-15}{5}  = -3
        \end{align*} 
    </p>
<p>
    We now have a formula for the slope of a line that works even if we do not have a graph.
</p>

<assemblage><title>Two-Point Slope Formula</title>
    <p>
        The slope of the line passing through the points <m>P_1 (x_1, y_1)</m> and <m>P_2 (x_2, y_2)</m> is given by
        <me>m = \frac{\Delta y}{\Delta x}= \frac{y_2 - y_1}{x_2 - x_1} 
            \text{, }~x_2 \ne x_1</me>
    </p>
</assemblage>

<example xml:id="example-two-point-slope">
    <p>Compute the slope of the line in <xref ref="fig-slope-formula" text="type-global" /> using the points <m>Q_1 (6, -3)</m> and <m>Q_2 (4, 3)</m>. </p>
    <solution><p>
        Substitute the coordinates of <m>Q_1</m> and <m>Q_2</m> into the slope formula to find
        <me>m = \frac{y_2 - y_1}{x_2 - x_1}= \frac{3 - (-3)}{4 - 6}
        = \frac{6}{-2}= -3</me>
        This value for the slope, <m>-3</m>, is the same value found above.
    </p></solution>
</example>

<exercise xml:id="exercise-two-point-slope"><statement>
    <ol label="a">
        <li><p>Find the slope of the line passing through the points <m>(2, -3)</m> and <m>( -2, -1)</m>.</p></li>
        <li><p>Sketch a graph of the line by hand.</p></li>
    </ol>
</statement>
<answer><p>
    <ol label="a" cols="2">
        <li><p><m>\dfrac{-1}{2} </m></p></li>
        <li><p><image source="images/fig-in-ex-ans-1-4-7"  width="80%"><description>line</description></image> </p></li>
    </ol>
</p></answer>
</exercise>

<p>
    It will also be useful to write the slope formula with function notation. Recall that <m>f (x)</m> is another symbol for <m>y</m>, and, in particular, that <m>y_1 = f (x_1)</m> and <m>y_2 = f (x_2)</m>. Thus, if <m>x_2 \ne x_1</m>, we have
</p>

<assemblage><title>Slope Formula in Function Notation</title><p>
    <me>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} 
            \text{, }~x_2 \ne x_1</me></p>
</assemblage>

<example xml:id="slopes-on-parabola">
    <sidebyside widths="50% 38%"><p>
        <xref ref="fig-slopes-on-parabola" text="type-global"/> shows a graph of 
        <me>f (x) = x_2 - 6x</me>
        <ol label="a">
            <li>Compute the slope of the line segment joining the points at <m>x = 1</m> and <m>x = 4</m>.</li>
            <li>Compute the slope of the line segment joining the points at <m>x = 2</m> and <m>x = 5</m>.</li>
        </ol>
    </p>
    <figure xml:id="fig-slopes-on-parabola"><caption></caption><image source="images/fig-slopes-on-parabola"><description>slopes on parabola</description></image></figure></sidebyside>
    <solution><p>
        <ol label="a">
            <li>We set <m>x_1 = \alert{1}</m> and <m>x_2 = \alert{4}</m> and find the function values at each point. 
            <me>f (x_1) = f (\alert{1}) = \alert{1}^2 - 6(\alert{1}) = -5</me>
            <me>f (x_2) = f (\alert{4}) = \alert{4}^2 - 6(\alert{4}) = -8</me>
            Then
            <me>m = \frac{f (x_2) - f (x_1)}{x_2 - x_1}
            =\frac{-8 - (-5)}{4 - 1}=\frac{-3}{3}= -1</me></li>

            <li>We set <m>x_1 = \alert{2}</m> and <m>x_2 = \alert{5}</m> and find the function values at each point. 
            <me>f (x_1) = f (\alert{2}) = \alert{2}^2 - 6(\alert{2}) = -8</me>
            <me>f (x_2) = f (\alert{5}) = \alert{5}^2 - 6(\alert{5}) = -5</me>
            Then
            <me>m = \frac{f (x_2) - f (x_1)}{x_2 - x_1}
            =\frac{-5 - (-8)}{5 - 2}=\frac{3}{3}= 1</me></li>
        </ol>
    </p></solution>
</example>

<p>Note that the graph of <m>f</m> is not a straight line and that the slope is not constant.</p>

<exercise xml:id="exercise-slope-of-secant"><statement>
    <sidebyside widths="50% 40%"><p><xref ref="fig-slope-on-curve" text="type-global"/> shows the graph of a function <m>f</m>.
    <ol label="a">
            <li>Find <m>f (3)</m> and <m>f (5)</m>. </li>
            <li>Compute the slope of the line segment joining the points at <m>x = 3</m> and <m>x = 5</m>.</li>
            <li>Write an expression for the slope of the line segment joining the points at <m>x = a</m> and <m>x = b</m>.</li>
    </ol></p>
    <figure xml:id="fig-slope-on-curve"><caption></caption><image source="images/fig-slope-on-curve"   width="80%"><description>increasing curve</description></image></figure></sidebyside>
    </statement>
<answer><p>
    <ol label="a" cols="2">
            <li><p><m>f (3) = 2, ~f (5) = 8</m></p> </li>
            <li><p><m>3</m></p></li>
            <li><p><m>\dfrac{f(b)-f(a)}{b-a} </m></p></li>
    </ol>
</p></answer>
    
</exercise>
</subsection>

<xi:include href="./summary-1-4.xml" /> <!-- summary  -->
<xi:include href="./section-1-4-exercises.xml" /> <!-- exercises  -->

</section> 
<!-- </book>  </mathbook> -->