chap2-summary.xml

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<section xml:id="chap2-summary" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Chapter Summary and Review</title>
<subsection><title>Key Concepts</title>
<p>
	<ol label="*1*">
		<li><p>We can solve equations of the form <m>a(px + q)^2 + r = 0</m> by extraction of roots.</p></li>
		<li><p>The formula for compound interest is <m>A = P (1 + r)^n</m>.</p></li>
		<li><p>Simple nonlinear equations can be solved by undoing the operations on the variable.</p></li>
		<li><p>
			<assemblage><title></title>
			<p>
			   The absolute value of <m>x</m> is defined by
			   <me>
			    \abs{x} =
			    \begin{cases}
			    x \amp \text{if } x\ge 0\\
			    -x  \amp \text{if } x\lt 0
			    \end{cases}
			    </me>
			</p>
			</assemblage>
		</p></li>
		<li><p>
			<assemblage><title></title>
			<p>
			   The absolute value has the following properties:</p><p>
			   <sidebyside><tabular halign="left">
			   	<row>
			   		<cell><m>\abs{a + b}\le\abs{a}+\abs{b} </m></cell>
			   		<cell>Triangle inequality</cell>
			   	</row>
			   	<row>
			   		<cell><m>\abs{ab}=\abs{a}\abs{b} </m></cell>
			   		<cell>Multiplicative property</cell>
			   	</row>
			   </tabular></sidebyside>
			</p>
			</assemblage>
		</p></li>
		<li><p>Many situations can be modeled by one of eight basic functions:</p><p>
			<sidebyside><tabular>
				<row>
					<cell><m>y=x</m></cell>
					<cell><m>y=\abs{x} </m></cell>
					<cell><m>y=x^2</m></cell>
					<cell><m>y=x^3</m></cell>
				</row>
				<row>
					<cell><m>y=\dfrac{1}{x} </m></cell>
					<cell><m>y=\dfrac{1}{x^2} </m></cell>
					<cell><m>y=\sqrt{x} </m></cell>
					<cell><m>y=\sqrt[3]{x} </m></cell>
				</row>
			</tabular></sidebyside>
		</p></li>
		<li><p>Functions can be defined piecewise, with different formulas on different intervals.</p></li>
		<li><p>
			<assemblage><title>Transformations of Functions</title><p>
                <ul>
                	<li><p>The graph of <m>y = f (x) + k</m> is <term>shifted vertically</term> compared to the graph of <m>y = f (x)</m>.</p></li>
                	<li><p>The graph of <m>y = f (x + h)</m> is <term>shifted horizontally</term> compared to the graph of <m>y = f (x)</m>.</p></li>
                	<li><p>The graph of <m>y = af(x)</m> is <term>stretched or compressed vertically</term> compared to the graph of y = f (x).</p></li>
                	<li><p>The graph of <m>y=-f (x)</m> is <term>reflected about the <m>x</m>-axis</term> compared to the graph of <m>y = f (x)</m>.</p></li>
                </ul>
			</p></assemblage>
		</p></li>
		<li><p>A nonlinear graph may be <term>concave up</term> or <term>concave down</term>. If a graph is concave up, its slope is increasing. If it is concave down, its slope is decreasing.</p></li>
		<li><p>The absolute value is used to model distance: The distance between two points <m>x</m> and <m>a</m> is given by <m>\abs{x - a}</m>.</p></li>
		<li><p>
			<assemblage><title>Absolute Value Equations and Inequalities</title><p>
                <ul>
                	<li><p>The equation <m>\abs{ax + b} = c ~~(c \gt 0)</m> is equivalent to
                	<me>ax + b = c \text{ or } ax + b =-c</me>
                	</p></li>
                	<li><p>If the solutions of the equation <m>\abs{ax + b} = c</m> are <m>r</m> and <m>s</m>, with <m>r\lt s</m>, then the solutions of <m>\abs{ax + b}\lt c</m> are <m>r\lt x\lt s</m>.</p></li>
                	<li><p>If the solutions of the equation <m>\abs{ax + b} = c</m> are <m>r</m> and <m>s</m>, with <m>r\lt s</m>, then the solutions of <m>\abs{ax + b}\gt c</m> are  <m>x\lt r</m> or <m>x\gt s</m>.</p></li>
                </ul>
			</p></assemblage>
		</p></li>
		<li><p>We can use absolute value notation to express error tolerances in measurements.</p></li>
		<li><p>The <term>domain</term> of a function is the set of permissible values for the input variable. The <term>range</term> is the set of function values (that is, values of the output variable) that correspond to the domain values.</p></li>
		<li><p>A relationship between two variables is a <term>function</term> if each element of the domain is paired with only one element of the range.</p></li>
		<li><p>We can identify the domain and range of a function from its graph. The domain is the set of input values of all points on the graph, and the range is the set of output values.</p></li>
		<li><p>If the domain of a function is not given as part of its definition, we assume that the domain is as large as possible. In many applications, however, we may restrict the domain and range of a function to suit the situation at hand.</p></li>
	</ol>.
</p>
</subsection>
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</section>
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