{"id":1861,"date":"2024-06-14T20:55:23","date_gmt":"2024-06-14T20:55:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1861"},"modified":"2025-08-14T01:12:33","modified_gmt":"2025-08-14T01:12:33","slug":"analysis-of-quadratic-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/analysis-of-quadratic-functions-learn-it-2\/","title":{"raw":"Analysis of Quadratic Functions: Learn It 2","rendered":"Analysis of Quadratic Functions: Learn It 2"},"content":{"raw":"

Finding the Maximum and Minimum Value of a Quadratic Function<\/h2>\r\nThere are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.\r\n\r\n
The vertex of a parabola is the highest (maximum) or lowest (minimum) point, depending on the direction the parabola opens.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Two Two parabola functions with key points and equations labeled[\/caption]\r\n\r\n<\/section>
A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased [latex]80[\/latex] feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Diagram Diagram of the garden and the backyard[\/caption]\r\n\r\nFind a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length [latex]L[\/latex]. Then, use the formula to answer: What dimensions should she make her garden to maximize the enclosed area?[reveal-answer q=\"958774\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"958774\"]Let\u2019s use a diagram to record the given information.Let [latex]L[\/latex] represents the length of the fence and [latex]W[\/latex] represents the width of the fence.\r\n\r\nWe know we have only [latex]80[\/latex] feet of fence available, and [latex]L+W+L=80[\/latex], or more simply, [latex]2L+W=80[\/latex].\r\n

This allows us to represent the width, [latex]W[\/latex], in terms of [latex]L[\/latex].<\/p>\r\n

[latex]W=80 - 2L[\/latex]<\/p>\r\nNow, we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width.\r\n

[latex]A=L \\cdot W=L \\cdot (80 - 2L)=80L - 2{L}^{2}[\/latex].<\/p>\r\nThis formula represents the area of the fence in terms of the variable length [latex]L[\/latex].\r\n\r\nThus, the function, written in general form, is[latex]A\\left(L\\right)=-2{L}^{2}+80L[\/latex].<\/strong>\r\n\r\nNow, to answer the question: What dimensions should she make her garden to maximize the enclosed area?\r\n\r\nThe quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area.\r\n\r\nBased on the function above, we have [latex]a=-2,b=80[\/latex], and [latex]c=0[\/latex].\r\n\r\nVertex:\r\n

[latex]h= -\\dfrac{b}{2a} = -\\dfrac{80}{2(-2)} = -\\dfrac{80}{-4} = 20[\/latex]<\/p>\r\n

and<\/p>\r\n

[latex]\\begin{align}k&=A\\left(20\\right) \\\\&=80\\left(20\\right)-2{\\left(20\\right)}^{2}\\\\&=800 \\end{align}[\/latex]<\/p>\r\nThus, the vertex is [latex](20, 800)[\/latex].\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"268\"]\"Graph Graph of a parabolic function with the vertex labeled[\/caption]\r\n\r\nThis problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function below.\r\n\r\nInterpretation of the vertex:<\/strong> The maximum value of the function is an area of [latex]800[\/latex] square feet, which occurs when [latex]L=20[\/latex] feet. When the shorter sides are [latex]L = 20[\/latex] feet, there is [latex]W = 80-2(20) = 40[\/latex] feet of fencing left for the longer side.\r\n\r\nSo, to maximize the area, she should enclose the garden so the two shorter sides have length [latex]20[\/latex] feet and the longer side parallel to the existing fence has length [latex]40[\/latex] feet.<\/strong>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>The problem we solved above is called a constrained optimization problem<\/strong>. We can optimize our desired outcome given a constraint, which in this case was a limited amount of fencing materials.\r\n\r\n

[ohm2_question hide_question_numbers=1]24488[\/ohm2_question]<\/section>","rendered":"

Finding the Maximum and Minimum Value of a Quadratic Function<\/h2>\n

There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.<\/p>\n

The vertex of a parabola is the highest (maximum) or lowest (minimum) point, depending on the direction the parabola opens.<\/p>\n
\"Two
Two parabola functions with key points and equations labeled<\/figcaption><\/figure>\n<\/section>\n
A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased [latex]80[\/latex] feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.<\/p>\n
\"Diagram
Diagram of the garden and the backyard<\/figcaption><\/figure>\n

Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length [latex]L[\/latex]. Then, use the formula to answer: What dimensions should she make her garden to maximize the enclosed area?<\/p>\n