{"id":3885,"date":"2021-05-20T18:29:54","date_gmt":"2021-05-20T18:29:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-defining-the-derivative\/"},"modified":"2021-07-03T18:48:17","modified_gmt":"2021-07-03T18:48:17","slug":"review-for-defining-the-derivative","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-defining-the-derivative\/","title":{"raw":"Skills Review for Defining the Derivative and the Derivative as a Function","rendered":"Skills Review for Defining the Derivative and the Derivative as a Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Given a function equation, find function values (outputs) for specified variables (inputs)&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Given a function equation, find function values (outputs) for specified variables (inputs)<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write the equation of a line using slope and a point on the line&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:13624051},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Write the equation of a line using slope and a point on the line<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Remove radicals from a multiple term denominator&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6915,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15987699],&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Calibri&quot;}\">Remove radicals from a multiple term denominator<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Simplify complex rational expressions\\r&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6915,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,16777215],&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Calibri&quot;}\">Simplify complex rational expressions <\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nThe first two sections of this module introduce you to the formal definition of a derivative which involves a large amount of algebra. Here we will review evaluating a function at variable inputs, including how to find a function's difference quotient. Since writing the equation of a tangent line is required in these sections, writing the equation of a line will also be reviewed. Finally, a refresher about how to rationalize and simplify complex rational expressions will be given.\r\n<h2>Evaluate Functions at Variable Inputs<\/h2>\r\nYou likely have plenty of experience evaluating functions at constants. Functions can also be evaluated for inputs that are variables or expressions. The process is the same, but the simplified answer will contain a variable. The following example shows you how to evaluate a function for a variable input.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example: Evaluating Functions at Variable Inputs<\/h3>\r\n<p id=\"fs-id1165134193005\">Evaluate [latex]f\\left(x\\right)={x}^{2}+3x - 4[\/latex] at<\/p>\r\n\r\n<ol id=\"fs-id1165137648008\">\r\n \t<li>[latex]2[\/latex]<\/li>\r\n \t<li>[latex]a[\/latex]<\/li>\r\n \t<li>[latex]a+h[\/latex]<\/li>\r\n \t<li>[latex]\\frac{f\\left(a+h\\right)-f\\left(a\\right)}{h}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"52497\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"52497\"]\r\n<p id=\"fs-id1165137936905\">Replace the [latex]x[\/latex]\u00a0in the function with each specified value.<\/p>\r\n\r\n<ol id=\"fs-id1165137778273\">\r\n \t<li>Because the input value is a number, 2, we can use algebra to simplify.\r\n<div id=\"fs-id1165135160774\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}f\\left(2\\right)&amp;={2}^{2}+3\\left(2\\right)-4 \\\\ &amp;=4+6 - 4 \\\\ &amp;=6 \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>In this case, the input value is a letter so we cannot simplify the answer any further.\r\n<div id=\"fs-id1165137638318\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div><\/li>\r\n \t<li>With an input value of [latex]a+h[\/latex], we must use the distributive property.\r\n<div id=\"fs-id1165137911654\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}f\\left(a+h\\right)&amp;={\\left(a+h\\right)}^{2}+3\\left(a+h\\right)-4 \\\\ &amp;={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that\r\n<div id=\"fs-id1165137638318\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(a+h\\right)={a}^{2}+2ah+{h}^{2}+3a+3h - 4[\/latex]<\/div>\r\nand we know that\r\n<div id=\"fs-id1165137638318\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div>\r\n<p id=\"fs-id1165137767461\">Now we combine the results and simplify.<\/p>\r\n\r\n<div id=\"fs-id1165137573884\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align} \\frac{f\\left(a+h\\right)-f\\left(a\\right)}{h}&amp;=\\frac{\\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\\right)-\\left({a}^{2}+3a - 4\\right)}{h} \\\\[1.5mm]&amp;=\\frac{2ah+{h}^{2}+3h}{h} \\\\[1.5mm]&amp;=\\frac{h\\left(2a+h+3\\right)}{h} &amp;&amp;\\text{Factor out }h. \\\\[1.5mm]&amp;=2a+h+3 &amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of evaluating functions for both constant and variable inputs.\r\n\r\nhttps:\/\/youtu.be\/_bi0B2zibOg\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExDetermineVariousFunctionOutputsForAQuadraticFunction_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \"Ex: Determine Various Function Outputs for a Quadratic Function\" here (opens in new window)<\/a>.\r\n<h2>Write the Equation of a Line<\/h2>\r\nTo write the equation of a line, the line's slope and a point the line goes through must be known. Perhaps the most familiar form of a linear equation is <strong>slope-intercept form<\/strong> written as [latex]y=mx+b[\/latex], where [latex]m=\\text{slope}[\/latex] and [latex]b=y\\text{-intercept}[\/latex]. Let us begin with the slope.\r\n\r\nOften, the starting point to writing the equation of a line is to use <strong>point-slope formula<\/strong>.\u00a0Given the slope and one point on a line, we can find the equation of the line using point-slope form shown below.\r\n<div style=\"text-align: center;\">[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/div>\r\nWe need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Point-Slope Formula<\/h3>\r\nGiven one point and the slope, using point-slope form will lead to the equation of a line:\r\n<div style=\"text-align: center;\">[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<div style=\"text-align: left;\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Equation of a Line Given the Slope and One Point<\/h3>\r\nWrite the equation of the line with slope [latex]m=-3[\/latex] and passing through the point [latex]\\left(4,8\\right)[\/latex]. Write the final equation in slope-intercept form.\r\n[reveal-answer q=\"201330\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"201330\"]\r\n\r\nUsing point-slope form, substitute [latex]-3[\/latex] for <em>m <\/em>and the point [latex]\\left(4,8\\right)[\/latex] for [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y-{y}_{1}=m\\left(x-{x}_{1}\\right)\\hfill \\\\ y - 8=-3\\left(x - 4\\right)\\hfill \\\\ y - 8=-3x+12\\hfill \\\\ y=-3x+20\\hfill \\end{array}[\/latex]<\/div>\r\n<div>\r\n<div>\r\n<h4>Analysis of the Solution<\/h4>\r\n<\/div>\r\n<div>\r\n\r\nNote that any point on the line can be used to find the equation. If done correctly, the same final equation will be obtained.\r\n\r\n<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven [latex]m=4[\/latex], find the equation of the line in slope-intercept form passing through the point [latex]\\left(2,5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"634647\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"634647\"]\r\n\r\n[latex]y=4x - 3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]110942[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Rationalize Radical Expressions<\/h2>\r\n<strong><em>(also in Module 2, Skills Review for The Limit Laws)<\/em><\/strong>\r\n\r\nWhen an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called <em>rationalizing the denominator<\/em>.\r\n\r\nWe know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by a form of 1 that will eliminate the radical.\r\n\r\nFor a denominator containing a binomial where at least one of the terms is a square root, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign in the middle of the binomial. If the denominator is [latex]a+b\\sqrt{c}[\/latex], then the conjugate is [latex]a-b\\sqrt{c}[\/latex].\r\n<div class=\"textbox\">\r\n<h3>How To: Given an expression with a Binomial containing a square root in the denominator, rationalize the denominator<\/h3>\r\n<ol>\r\n \t<li>Find the conjugate of the denominator.<\/li>\r\n \t<li>Multiply the numerator and denominator by the conjugate.<\/li>\r\n \t<li>Use the distributive property.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Rationalizing a Denominator with a binomial Containing a square root<\/h3>\r\nRationalize [latex]\\dfrac{4}{1+\\sqrt{5}}[\/latex].\r\n\r\n[reveal-answer q=\"726340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"726340\"]\r\n\r\nBegin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of [latex]1+\\sqrt{5}[\/latex] is [latex]1-\\sqrt{5}[\/latex]. Then multiply the fraction by [latex]\\dfrac{1-\\sqrt{5}}{1-\\sqrt{5}}[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{1+\\sqrt{5}}\\cdot \\frac{1-\\sqrt{5}}{1-\\sqrt{5}} &amp;= \\frac{4 - 4\\sqrt{5}}{-4} &amp;&amp; \\text{Use the distributive property}. \\\\ &amp;=\\sqrt{5}-1 &amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite [latex]\\dfrac{7}{2+\\sqrt{3}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"132932\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"132932\"]\r\n\r\n[latex]14 - 7\\sqrt{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]3441[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=vINRIRgeKqU&amp;feature=youtu.be\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExRationalizeTheDenominatorOfARadicalExpressionConjugate_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \"Ex: Rationalize the Denominator of a Radical Expression - Conjugate\" here (opens in new window)<\/a>.\r\n<h2>Simplify Complex Rational Expressions<\/h2>\r\n<strong><em>(also in Module 2, Skills Review for The Limit Laws)<\/em><\/strong>\r\n\r\n<\/div>\r\nA complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\\dfrac{a}{\\dfrac{1}{b}+c}[\/latex] can be simplified by rewriting the numerator as the fraction [latex]\\dfrac{a}{1}[\/latex] and combining the expressions in the denominator as [latex]\\dfrac{1+bc}{b}[\/latex]. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\\dfrac{a}{1}\\cdot \\dfrac{b}{1+bc}[\/latex] which is equal to [latex]\\dfrac{ab}{1+bc}[\/latex].\r\n<div class=\"textbox\">\r\n<h3>How To: Given a complex rational expression, simplify it<\/h3>\r\n<ol>\r\n \t<li>Combine the expressions in the numerator into a single rational expression by adding or subtracting.<\/li>\r\n \t<li>Combine the expressions in the denominator into a single rational expression by adding or subtracting.<\/li>\r\n \t<li>Rewrite as the numerator divided by the denominator.<\/li>\r\n \t<li>Rewrite as multiplication.<\/li>\r\n \t<li>Multiply.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying Complex Rational Expressions<\/h3>\r\nSimplify: [latex]\\dfrac{y+\\dfrac{1}{x}}{\\dfrac{x}{y}}[\/latex] .\r\n\r\n[reveal-answer q=\"967019\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"967019\"]\r\n\r\nBegin by combining the expressions in the numerator into one expression.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}y\\cdot \\dfrac{x}{x}+\\dfrac{1}{x}\\hfill &amp; \\text{Multiply by }\\dfrac{x}{x}\\text{to get LCD as denominator}.\\hfill \\\\ \\dfrac{xy}{x}+\\dfrac{1}{x}\\hfill &amp; \\\\ \\dfrac{xy+1}{x}\\hfill &amp; \\text{Add numerators}.\\hfill \\end{array}[\/latex]<\/div>\r\nNow the numerator is a single rational expression and the denominator is a single rational expression.\r\n<div style=\"text-align: center;\">[latex]\\dfrac{\\dfrac{xy+1}{x}}{\\dfrac{x}{y}}[\/latex]<\/div>\r\nWe can rewrite this as division and then multiplication.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\dfrac{xy+1}{x}\\div \\dfrac{x}{y}\\hfill &amp; \\\\ \\dfrac{xy+1}{x}\\cdot \\dfrac{y}{x}\\hfill &amp; \\text{Rewrite as multiplication}\\text{.}\\hfill \\\\ \\dfrac{y\\left(xy+1\\right)}{{x}^{2}}\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]3078-3080-59554[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Given a function equation, find function values (outputs) for specified variables (inputs)&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Given a function equation, find function values (outputs) for specified variables (inputs)<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write the equation of a line using slope and a point on the line&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:13624051},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Write the equation of a line using slope and a point on the line<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Remove radicals from a multiple term denominator&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6915,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15987699],&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Calibri&quot;}\">Remove radicals from a multiple term denominator<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Simplify complex rational expressions\\r&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6915,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,16777215],&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Calibri&quot;}\">Simplify complex rational expressions <\/span><\/li>\n<\/ul>\n<\/div>\n<p>The first two sections of this module introduce you to the formal definition of a derivative which involves a large amount of algebra. Here we will review evaluating a function at variable inputs, including how to find a function&#8217;s difference quotient. Since writing the equation of a tangent line is required in these sections, writing the equation of a line will also be reviewed. Finally, a refresher about how to rationalize and simplify complex rational expressions will be given.<\/p>\n<h2>Evaluate Functions at Variable Inputs<\/h2>\n<p>You likely have plenty of experience evaluating functions at constants. Functions can also be evaluated for inputs that are variables or expressions. The process is the same, but the simplified answer will contain a variable. The following example shows you how to evaluate a function for a variable input.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example: Evaluating Functions at Variable Inputs<\/h3>\n<p id=\"fs-id1165134193005\">Evaluate [latex]f\\left(x\\right)={x}^{2}+3x - 4[\/latex] at<\/p>\n<ol id=\"fs-id1165137648008\">\n<li>[latex]2[\/latex]<\/li>\n<li>[latex]a[\/latex]<\/li>\n<li>[latex]a+h[\/latex]<\/li>\n<li>[latex]\\frac{f\\left(a+h\\right)-f\\left(a\\right)}{h}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q52497\">Show Solution<\/span><\/p>\n<div id=\"q52497\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137936905\">Replace the [latex]x[\/latex]\u00a0in the function with each specified value.<\/p>\n<ol id=\"fs-id1165137778273\">\n<li>Because the input value is a number, 2, we can use algebra to simplify.\n<div id=\"fs-id1165135160774\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}f\\left(2\\right)&={2}^{2}+3\\left(2\\right)-4 \\\\ &=4+6 - 4 \\\\ &=6 \\end{align}[\/latex]<\/div>\n<\/li>\n<li>In this case, the input value is a letter so we cannot simplify the answer any further.\n<div id=\"fs-id1165137638318\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div>\n<\/li>\n<li>With an input value of [latex]a+h[\/latex], we must use the distributive property.\n<div id=\"fs-id1165137911654\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}f\\left(a+h\\right)&={\\left(a+h\\right)}^{2}+3\\left(a+h\\right)-4 \\\\ &={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \\end{align}[\/latex]<\/div>\n<\/li>\n<li>In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that\n<div id=\"fs-id1165137638318\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(a+h\\right)={a}^{2}+2ah+{h}^{2}+3a+3h - 4[\/latex]<\/div>\n<p>and we know that<\/p>\n<div id=\"fs-id1165137638318\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div>\n<p id=\"fs-id1165137767461\">Now we combine the results and simplify.<\/p>\n<div id=\"fs-id1165137573884\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align} \\frac{f\\left(a+h\\right)-f\\left(a\\right)}{h}&=\\frac{\\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\\right)-\\left({a}^{2}+3a - 4\\right)}{h} \\\\[1.5mm]&=\\frac{2ah+{h}^{2}+3h}{h} \\\\[1.5mm]&=\\frac{h\\left(2a+h+3\\right)}{h} &&\\text{Factor out }h. \\\\[1.5mm]&=2a+h+3 && \\text{Simplify}. \\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of evaluating functions for both constant and variable inputs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Determine Various Function Outputs for a Quadratic Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_bi0B2zibOg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExDetermineVariousFunctionOutputsForAQuadraticFunction_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;Ex: Determine Various Function Outputs for a Quadratic Function&#8221; here (opens in new window)<\/a>.<\/p>\n<h2>Write the Equation of a Line<\/h2>\n<p>To write the equation of a line, the line&#8217;s slope and a point the line goes through must be known. Perhaps the most familiar form of a linear equation is <strong>slope-intercept form<\/strong> written as [latex]y=mx+b[\/latex], where [latex]m=\\text{slope}[\/latex] and [latex]b=y\\text{-intercept}[\/latex]. Let us begin with the slope.<\/p>\n<p>Often, the starting point to writing the equation of a line is to use <strong>point-slope formula<\/strong>.\u00a0Given the slope and one point on a line, we can find the equation of the line using point-slope form shown below.<\/p>\n<div style=\"text-align: center;\">[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/div>\n<p>We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Point-Slope Formula<\/h3>\n<p>Given one point and the slope, using point-slope form will lead to the equation of a line:<\/p>\n<div style=\"text-align: center;\">[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/div>\n<\/div>\n<div style=\"text-align: left;\">\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Equation of a Line Given the Slope and One Point<\/h3>\n<p>Write the equation of the line with slope [latex]m=-3[\/latex] and passing through the point [latex]\\left(4,8\\right)[\/latex]. Write the final equation in slope-intercept form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q201330\">Show Solution<\/span><\/p>\n<div id=\"q201330\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using point-slope form, substitute [latex]-3[\/latex] for <em>m <\/em>and the point [latex]\\left(4,8\\right)[\/latex] for [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y-{y}_{1}=m\\left(x-{x}_{1}\\right)\\hfill \\\\ y - 8=-3\\left(x - 4\\right)\\hfill \\\\ y - 8=-3x+12\\hfill \\\\ y=-3x+20\\hfill \\end{array}[\/latex]<\/div>\n<div>\n<div>\n<h4>Analysis of the Solution<\/h4>\n<\/div>\n<div>\n<p>Note that any point on the line can be used to find the equation. If done correctly, the same final equation will be obtained.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given [latex]m=4[\/latex], find the equation of the line in slope-intercept form passing through the point [latex]\\left(2,5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q634647\">Show Solution<\/span><\/p>\n<div id=\"q634647\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]y=4x - 3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm110942\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110942&theme=oea&iframe_resize_id=ohm110942&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Rationalize Radical Expressions<\/h2>\n<p><strong><em>(also in Module 2, Skills Review for The Limit Laws)<\/em><\/strong><\/p>\n<p>When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called <em>rationalizing the denominator<\/em>.<\/p>\n<p>We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by a form of 1 that will eliminate the radical.<\/p>\n<p>For a denominator containing a binomial where at least one of the terms is a square root, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign in the middle of the binomial. If the denominator is [latex]a+b\\sqrt{c}[\/latex], then the conjugate is [latex]a-b\\sqrt{c}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given an expression with a Binomial containing a square root in the denominator, rationalize the denominator<\/h3>\n<ol>\n<li>Find the conjugate of the denominator.<\/li>\n<li>Multiply the numerator and denominator by the conjugate.<\/li>\n<li>Use the distributive property.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Rationalizing a Denominator with a binomial Containing a square root<\/h3>\n<p>Rationalize [latex]\\dfrac{4}{1+\\sqrt{5}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q726340\">Show Solution<\/span><\/p>\n<div id=\"q726340\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of [latex]1+\\sqrt{5}[\/latex] is [latex]1-\\sqrt{5}[\/latex]. Then multiply the fraction by [latex]\\dfrac{1-\\sqrt{5}}{1-\\sqrt{5}}[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{1+\\sqrt{5}}\\cdot \\frac{1-\\sqrt{5}}{1-\\sqrt{5}} &= \\frac{4 - 4\\sqrt{5}}{-4} && \\text{Use the distributive property}. \\\\ &=\\sqrt{5}-1 && \\text{Simplify}. \\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]\\dfrac{7}{2+\\sqrt{3}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q132932\">Show Solution<\/span><\/p>\n<div id=\"q132932\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]14 - 7\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3441\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3441&theme=oea&iframe_resize_id=ohm3441&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Rationalize the Denominator of a Radical Expression - Conjugate\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vINRIRgeKqU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExRationalizeTheDenominatorOfARadicalExpressionConjugate_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;Ex: Rationalize the Denominator of a Radical Expression &#8211; Conjugate&#8221; here (opens in new window)<\/a>.<\/p>\n<h2>Simplify Complex Rational Expressions<\/h2>\n<p><strong><em>(also in Module 2, Skills Review for The Limit Laws)<\/em><\/strong><\/p>\n<\/div>\n<p>A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\\dfrac{a}{\\dfrac{1}{b}+c}[\/latex] can be simplified by rewriting the numerator as the fraction [latex]\\dfrac{a}{1}[\/latex] and combining the expressions in the denominator as [latex]\\dfrac{1+bc}{b}[\/latex]. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\\dfrac{a}{1}\\cdot \\dfrac{b}{1+bc}[\/latex] which is equal to [latex]\\dfrac{ab}{1+bc}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a complex rational expression, simplify it<\/h3>\n<ol>\n<li>Combine the expressions in the numerator into a single rational expression by adding or subtracting.<\/li>\n<li>Combine the expressions in the denominator into a single rational expression by adding or subtracting.<\/li>\n<li>Rewrite as the numerator divided by the denominator.<\/li>\n<li>Rewrite as multiplication.<\/li>\n<li>Multiply.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Complex Rational Expressions<\/h3>\n<p>Simplify: [latex]\\dfrac{y+\\dfrac{1}{x}}{\\dfrac{x}{y}}[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q967019\">Show Solution<\/span><\/p>\n<div id=\"q967019\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by combining the expressions in the numerator into one expression.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}y\\cdot \\dfrac{x}{x}+\\dfrac{1}{x}\\hfill & \\text{Multiply by }\\dfrac{x}{x}\\text{to get LCD as denominator}.\\hfill \\\\ \\dfrac{xy}{x}+\\dfrac{1}{x}\\hfill & \\\\ \\dfrac{xy+1}{x}\\hfill & \\text{Add numerators}.\\hfill \\end{array}[\/latex]<\/div>\n<p>Now the numerator is a single rational expression and the denominator is a single rational expression.<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{\\dfrac{xy+1}{x}}{\\dfrac{x}{y}}[\/latex]<\/div>\n<p>We can rewrite this as division and then multiplication.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\dfrac{xy+1}{x}\\div \\dfrac{x}{y}\\hfill & \\\\ \\dfrac{xy+1}{x}\\cdot \\dfrac{y}{x}\\hfill & \\text{Rewrite as multiplication}\\text{.}\\hfill \\\\ \\dfrac{y\\left(xy+1\\right)}{{x}^{2}}\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3078\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3078-3080-59554&theme=oea&iframe_resize_id=ohm3078&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3885\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision . <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t 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