chap6-summary.xml

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<section xml:id="chap6-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>A <term>quadratic</term> function has the form <m>f (x) = ax2 + bx + c</m>, where <m>a</m>, <m>b</m>, and <m>c</m> are constants and <m>a</m> is not equal to zero.
		</p></li>
		<li><p>
			<xref ref="assemblage-zero-factor-principle"><term>Zero-factor principle</term></xref>
		</p></li>
		<li><p>
			The <m>x</m>-intercepts of the graph of <m>y = f (x)</m> are the solutions of the equation <m>f (x) = 0</m>.
		</p></li>
		<li><p>
			A quadratic equation written as <m>ax^2 +bx+c=0</m> is in <term>standard form</term>.</p>
			<p>A quadratic equation written as <m>a(x - r_1 )(x - r_2)=0</m> is in <term>factored form</term>.
		</p></li>
		<li><p>
			<xref ref="assemblage-solve-quadratic-by-factoring">To Solve a Quadratic Equation by Factoring</xref>
		</p></li>
		<li><p>
			Every quadratic equation has two solutions, which may be the same.
		</p></li>
		<li><p>
			The value of the constant <m>a</m> in the factored form of a quadratic equation does not affect the solutions.
		</p></li>
		<li><p>
			Each solution of a quadratic equation corresponds to a factor in the factored form.
		</p></li>
		<li><p>An equation is called <term>quadratic in form</term> if we can use a substitution to write it as <m>au^2 + bu + c = 0</m>, where <m>u</m> stands for an algebraic expression.</p></li>
		<li><p>The square of the binomial is a <term>quadratic trinomial</term>,
		<me>(x+p)^2 =x^2 +2px+p^2</me>
		</p></li>
		<li><p>
			<xref ref="assemblage-completing-the-square">To Solve a Quadratic Equation by Completing the Square</xref>
		</p></li>
		<li><p><xref ref="assemblage-quadratic-formula">The Quadratic Formula</xref></p></li>
		<li><p>We have four methods for solving quadratic equations: extracting roots, factoring, completing the square, and using the quadratic formula. The first two methods are faster, but they do not work on all equations. The last two methods work on any quadratic equation.</p></li>
		<li><p>The graph of a quadratic function <m>f (x) = ax^2 + bx + c</m> is called a <term>parabola</term>. The values of the constants <m>a</m>, <m>b</m>, and <m>c</m> determine the location and orientation of the parabola.</p></li>
		<li><p>For the graph of <m>y = ax^2 + bx + c</m>, the <m>x</m>-coordinate of the vertex is <m>x_v = \dfrac{-b}{2a}</m>.</p>
		<p>To find the <m>y</m>-coordinate of the vertex, we substitute <m>x_v</m> into the formula for the parabola.</p></li>
		<li><p>The graph of the quadratic function <m>y = ax^2 + bx + c</m> may have two, one, or no <m>x</m>-intercepts, according to the number of distinct real-valued solutions of the equation <m>ax^2 + bx + c = 0</m>.</p></li>
		<li><p>
			<xref ref="assemblage-discriminant">The Discriminant</xref>
		</p></li>
		<li><p>
			<xref ref="assemblage-graph-quadratic-function">To graph the quadratic function <m>y = ax^2 + bx + c</m></xref>
		</p></li>
		<li><p>Quadratic models may arise as the product of two variables.</p></li>
		<li><p>The maximum or minimum of a quadratic function occurs at the vertex.</p></li>
		<li><p>
			<xref ref="assemblage-vertex-form-quadratic-function">Vertex Form for a Quadratic Function</xref>
		</p></li>
		<li><p>We can convert a quadratic equation to vertex form by completing the square.</p></li>
		<li><p>We can graph a quadratic equation in vertex form using transformations.</p></li>
		<li><p>A <m>2\times 2</m> system involving quadratic equations may have one, two, or no solutions.</p></li>
		<li><p>We can use a graphical technique to solve quadratic inequalities.</p></li>
		<li><p>
			<xref ref="assemblage-solve-quadratic-inequality-algebraically">To Solve a Quadratic Inequality Algebraically</xref>
		</p></li>
		<li><p>We need three points to determine a parabola.</p></li>
		<li><p>We can use the method of elimination to find the equation of a parabola through three points.</p></li>
		<li><p>If we know the vertex of a parabola, we need only one other point to find its equation.</p></li>
		<li><p>We can use quadratic regression to fit a parabola to a collection of data points.</p></li>
	</ol>.
</p>
</subsection>
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</section>
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