{"id":4105,"date":"2022-04-14T18:15:54","date_gmt":"2022-04-14T18:15:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-second-order-linear-equations\/"},"modified":"2022-11-09T16:47:31","modified_gmt":"2022-11-09T16:47:31","slug":"skills-review-for-second-order-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-second-order-linear-equations\/","title":{"raw":"Skills Review for Second-Order Linear Equations","rendered":"Skills Review for Second-Order Linear Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Given a function equation, find function values (outputs) for specified variables (inputs)<\/li>\r\n \t<li>Apply the Quadratic Formula<\/li>\r\n \t<li>Write function equations using given conditions<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Second-Order Linear Equations section, we will look at differential equations that contain second derivatives and a dependent variable that is not raised to any powers itself. Here we will review how to plug variable inputs into equations, use the Quadratic Formula, and write function equations using given conditions.\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<h2>Apply the Quadratic Formula<\/h2>\r\nThe\u00a0<strong>quadratic formula <\/strong>is\u00a0a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Quadratic Formula<\/h3>\r\nWritten in standard form, [latex]a{x}^{2}+bx+c=0[\/latex], any quadratic equation can be solved using the <strong>quadratic formula<\/strong>:\r\n<div style=\"text-align: center;\">[latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/div>\r\nwhere <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> are real numbers and [latex]a\\ne 0[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a quadratic equation, solve it using the quadratic formula<\/h3>\r\n<ol>\r\n \t<li>Make sure the equation is in standard form: [latex]a{x}^{2}+bx+c=0[\/latex].<\/li>\r\n \t<li>Make note of the values of the coefficients and constant term, [latex]a,b[\/latex], and [latex]c[\/latex].<\/li>\r\n \t<li>Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.<\/li>\r\n \t<li>Calculate and solve.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example : Solve A Quadratic Equation Using the Quadratic Formula<\/h3>\r\nSolve the quadratic equation: [latex]{x}^{2}+5x+1=0[\/latex].\r\n\r\n[reveal-answer q=\"641400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"641400\"]\r\n\r\nIdentify the coefficients: [latex]a=1,b=5,c=1[\/latex]. Then use the quadratic formula.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x\\hfill&amp;=\\frac{-\\left(5\\right)\\pm \\sqrt{{\\left(5\\right)}^{2}-4\\left(1\\right)\\left(1\\right)}}{2\\left(1\\right)}\\hfill \\\\ \\hfill&amp;=\\frac{-5\\pm \\sqrt{25 - 4}}{2}\\hfill \\\\ \\hfill&amp;=\\frac{-5\\pm \\sqrt{21}}{2}\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a Quadratic Equation with the Quadratic Formula<\/h3>\r\nUse the quadratic formula to solve [latex]{x}^{2}+x+2=0[\/latex].\r\n\r\n[reveal-answer q=\"688902\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688902\"]\r\n\r\nFirst, we identify the coefficients: [latex]a=1,b=1[\/latex], and [latex]c=2[\/latex].\r\n\r\nSubstitute these values into the quadratic formula.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x\\hfill&amp;=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\\\\\hfill&amp;=\\dfrac{-\\left(1\\right)\\pm \\sqrt{{\\left(1\\right)}^{2}-\\left(4\\right)\\cdot \\left(1\\right)\\cdot \\left(2\\right)}}{2\\cdot 1}\\hfill \\\\\\hfill&amp;=\\dfrac{-1\\pm \\sqrt{1 - 8}}{2}\\hfill \\\\ \\hfill&amp;=\\dfrac{-1\\pm \\sqrt{-7}}{2}\\hfill \\end{array}[\/latex]<\/div>\r\n<div><\/div>\r\nThe solutions to the equation are <em>unreal <\/em>because the square root of a negative number does not exist in the real numbers. If asked to find all <em>real<\/em> solutions, we would indicate that they do not exist by writing\u00a0<em>DNE<\/em>.\r\n\r\nWe may also express unreal solutions as complex numbers by writing\r\n\r\n[latex]x=\\dfrac{-1\\pm i\\sqrt{7}}{2}[\/latex]\r\n\r\nor\r\n\r\n[latex]x=-\\dfrac{1}{2}+\\dfrac{\\sqrt{7}}{2}i[\/latex] and [latex]x=-\\dfrac{1}{2}-\\dfrac{\\sqrt{7}}{2}i[\/latex]\u00a0,\u00a0where [latex]\\sqrt{-1}[\/latex] is expressed as [latex]i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the quadratic equation using the quadratic formula: [latex]9{x}^{2}+3x - 2=0[\/latex].\r\n\r\n[reveal-answer q=\"232269\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232269\"]\r\n\r\n[latex]x=-\\frac{2}{3}[\/latex], [latex]x=\\frac{1}{3}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4014&amp;theme=oea&amp;iframe_resize_id=mom3[\/embed]\r\n\r\n<\/div>\r\n<h2>Write Function Equations Using Given Conditions<\/h2>\r\n<strong><em>(also in Module 3, Skills Review for Motion in Space)<\/em><\/strong>\r\nSometimes, to find a missing value in a function equation, you will be given an input of the function and the corresponding output. You will then plug this input and output into the function equation and find the missing value.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing a Function Equation from given conditions<\/h3>\r\nGiven [latex]f(2)=-1[\/latex], find the unknown value c in the function equation [latex]f(x)=3x^3-4x^2-x+c[\/latex].\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"338564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"338564\"]\r\n\r\nTo find c, we use the fact that [latex]f(2)=-1[\/latex], that is, the function's value is -1 when [latex]x=2[\/latex].\r\n\r\n[latex]\\begin{array}{l}-1=3(2)^3-4(2)^2-2+c\\hfill \\\\ -1=3(8)-4(4)-2+c\\hfill \\\\ -1=24-16-2+c\\hfill \\\\ -1=6+c\\hfill \\\\ -7=c \\end{array}[\/latex]\r\n\r\nThe function equation is\u00a0[latex]f(x)=3x^3-4x^2-x-7[\/latex].\r\n\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]f(1)=5[\/latex], find the unknown value c in the function equation [latex]f(x)=-2x^2+3x+c[\/latex].\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"338565\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"338565\"]\r\n\r\nThe function equation is [latex]f(x)=-2x^2+3x+4[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Given a function equation, find function values (outputs) for specified variables (inputs)<\/li>\n<li>Apply the Quadratic Formula<\/li>\n<li>Write function equations using given conditions<\/li>\n<\/ul>\n<\/div>\n<p>In the Second-Order Linear Equations section, we will look at differential equations that contain second derivatives and a dependent variable that is not raised to any powers itself. Here we will review how to plug variable inputs into equations, use the Quadratic Formula, and write function equations using given conditions.<\/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<h2>Apply the Quadratic Formula<\/h2>\n<p>The\u00a0<strong>quadratic formula <\/strong>is\u00a0a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Quadratic Formula<\/h3>\n<p>Written in standard form, [latex]a{x}^{2}+bx+c=0[\/latex], any quadratic equation can be solved using the <strong>quadratic formula<\/strong>:<\/p>\n<div style=\"text-align: center;\">[latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/div>\n<p>where <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> are real numbers and [latex]a\\ne 0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a quadratic equation, solve it using the quadratic formula<\/h3>\n<ol>\n<li>Make sure the equation is in standard form: [latex]a{x}^{2}+bx+c=0[\/latex].<\/li>\n<li>Make note of the values of the coefficients and constant term, [latex]a,b[\/latex], and [latex]c[\/latex].<\/li>\n<li>Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.<\/li>\n<li>Calculate and solve.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example : Solve A Quadratic Equation Using the Quadratic Formula<\/h3>\n<p>Solve the quadratic equation: [latex]{x}^{2}+5x+1=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q641400\">Show Solution<\/span><\/p>\n<div id=\"q641400\" class=\"hidden-answer\" style=\"display: none\">\n<p>Identify the coefficients: [latex]a=1,b=5,c=1[\/latex]. Then use the quadratic formula.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x\\hfill&=\\frac{-\\left(5\\right)\\pm \\sqrt{{\\left(5\\right)}^{2}-4\\left(1\\right)\\left(1\\right)}}{2\\left(1\\right)}\\hfill \\\\ \\hfill&=\\frac{-5\\pm \\sqrt{25 - 4}}{2}\\hfill \\\\ \\hfill&=\\frac{-5\\pm \\sqrt{21}}{2}\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Quadratic Equation with the Quadratic Formula<\/h3>\n<p>Use the quadratic formula to solve [latex]{x}^{2}+x+2=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q688902\">Show Solution<\/span><\/p>\n<div id=\"q688902\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we identify the coefficients: [latex]a=1,b=1[\/latex], and [latex]c=2[\/latex].<\/p>\n<p>Substitute these values into the quadratic formula.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x\\hfill&=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\\\\\hfill&=\\dfrac{-\\left(1\\right)\\pm \\sqrt{{\\left(1\\right)}^{2}-\\left(4\\right)\\cdot \\left(1\\right)\\cdot \\left(2\\right)}}{2\\cdot 1}\\hfill \\\\\\hfill&=\\dfrac{-1\\pm \\sqrt{1 - 8}}{2}\\hfill \\\\ \\hfill&=\\dfrac{-1\\pm \\sqrt{-7}}{2}\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<p>The solutions to the equation are <em>unreal <\/em>because the square root of a negative number does not exist in the real numbers. If asked to find all <em>real<\/em> solutions, we would indicate that they do not exist by writing\u00a0<em>DNE<\/em>.<\/p>\n<p>We may also express unreal solutions as complex numbers by writing<\/p>\n<p>[latex]x=\\dfrac{-1\\pm i\\sqrt{7}}{2}[\/latex]<\/p>\n<p>or<\/p>\n<p>[latex]x=-\\dfrac{1}{2}+\\dfrac{\\sqrt{7}}{2}i[\/latex] and [latex]x=-\\dfrac{1}{2}-\\dfrac{\\sqrt{7}}{2}i[\/latex]\u00a0,\u00a0where [latex]\\sqrt{-1}[\/latex] is expressed as [latex]i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the quadratic equation using the quadratic formula: [latex]9{x}^{2}+3x - 2=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q232269\">Show Solution<\/span><\/p>\n<div id=\"q232269\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=-\\frac{2}{3}[\/latex], [latex]x=\\frac{1}{3}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm4014\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4014&#38;theme=oea&#38;iframe_resize_id=ohm4014&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Write Function Equations Using Given Conditions<\/h2>\n<p><strong><em>(also in Module 3, Skills Review for Motion in Space)<\/em><\/strong><br \/>\nSometimes, to find a missing value in a function equation, you will be given an input of the function and the corresponding output. You will then plug this input and output into the function equation and find the missing value.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Function Equation from given conditions<\/h3>\n<p>Given [latex]f(2)=-1[\/latex], find the unknown value c in the function equation [latex]f(x)=3x^3-4x^2-x+c[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q338564\">Show Solution<\/span><\/p>\n<div id=\"q338564\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find c, we use the fact that [latex]f(2)=-1[\/latex], that is, the function&#8217;s value is -1 when [latex]x=2[\/latex].<\/p>\n<p>[latex]\\begin{array}{l}-1=3(2)^3-4(2)^2-2+c\\hfill \\\\ -1=3(8)-4(4)-2+c\\hfill \\\\ -1=24-16-2+c\\hfill \\\\ -1=6+c\\hfill \\\\ -7=c \\end{array}[\/latex]<\/p>\n<p>The function equation is\u00a0[latex]f(x)=3x^3-4x^2-x-7[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given [latex]f(1)=5[\/latex], find the unknown value c in the function equation [latex]f(x)=-2x^2+3x+c[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q338565\">Show Solution<\/span><\/p>\n<div id=\"q338565\" class=\"hidden-answer\" style=\"display: none\">\n<p>The function equation is [latex]f(x)=-2x^2+3x+4[\/latex].<\/p>\n<\/div>\n<\/div>\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-4105\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/calculus1\/\">https:\/\/courses.lumenlearning.com\/calculus1\/<\/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>Calculus Volume 2. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/calculus2\/\">https:\/\/courses.lumenlearning.com\/calculus2\/<\/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 <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"\",\"organization\":\"Lumen 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