IM Alg1.7.23 Lesson: Using Quadratic Expressions in Vertex Form to Solve Problems

Here are graphs that represent two functions, f and g, defined by:
[table][tr][td][math]f(x)=(x-4)^2+1[/math][br][br][br][math]g(x)=\text{-}(x-12)^2+7[/math][/td][td][img]data:image/png;base64,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[/img][/td][/tr][/table][br][size=100][size=150][math]f\left(1\right)[/math] can be expressed in words as “the value of [math]f[/math] when [math]x[/math] is 1.” Find or compute: [/size][/size][br][br]the value of [math]f[/math] when [math]x[/math] is 1
[math]f\left(3\right)[/math]
[math]f\left(10\right)[/math]
[size=150]Can you find an value that would make [math]f\left(x\right)[/math]:[/size][br][br]Less than 1?
Greater than 10,000?
[size=150][math]g\left(9\right)[/math] can be expressed in words as “the value of [math]g[/math] when [math]x[/math] is 9.” Find or compute:[/size][br][br] the value of [math]g[/math] when [math]x[/math] is 9
[math]g\left(13\right)[/math]
[math]g\left(2\right)[/math]
[size=150]Can you find an [math]x[/math] value that would make [math]g\left(x\right)[/math]:[/size][br][br]Greater than 7?
Less than -10,000?
[size=150]The graph that represents [math]p\left(x\right)=\left(x-8\right)^2+1[/math] has its vertex at [math]\left(8,1\right)[/math]. Here is one way to show, without graphing, that [math]\left(8,1\right)[/math] corresponds to the [i]minimum[/i] value of [math]p[/math].[br][br][list][*]When [math]x=8[/math], the value of [math]\left(x-8\right)^2[/math] is 0, because [math]\left(8-8\right)^2=0^2=0[/math].[/*][*]Squaring any number always results in a positive number, so when [math]x[/math] is any value other than 8, [math]\left(x-8\right)[/math] will be a number other than 0, and when squared, [math]\left(x-8\right)^2[/math] will be positive.[/*][*]Any positive number is greater than 0, so when [math]x\ne8[/math], the value of [math]\left(x-8\right)^2[/math] will be greater than when [math]x=8[/math]. In other words, [math]p[/math] has the least value when [math]x=8[/math].[/*][/list][/size]Use similar reasoning to explain why the point [math]\left(4,1\right)[/math] corresponds to the [i]maximum[/i] value of [math]q[/math], defined by [math]q\left(x\right)=-2\left(x-4\right)^2+1[/math].
Here are some quadratic functions, and the coordinates of the vertex of the graph of each. Determine if the vertex corresponds to the maximum or the minimum value of the function. Be prepared to explain how you know.
[size=150]Here is a portion of the graph of function [math]q[/math], defined by [math]q\left(x\right)=-x^2+14x-40[/math].[br][table][tr][td][br][img]data:image/png;base64,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[/img][/td][td][br][math]ABCD[/math] is a rectangle. Points [math]A[/math] and [math]B[/math] coincide with [br]the [math]x[/math]-intercepts of the graph, and segment [math]CD[/math] just [br]touches the vertex of the graph.[br][/td][/tr][/table][/size][br]Find the area of [math]ABCD[/math]. Show your reasoning.[br]
[size=150]A function [math]A[/math], defined by [math]p(600-75p)[/math], describes the revenue collected from the sales of tickets for Performance A, a musical.[br][br]The graph represents a function [math]B[/math] that models the revenue collected from the sales of tickets for Performance B, a Shakespearean comedy.[br][img]data:image/png;base64,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[/img][br]In both functions, [math]p[/math] represents the price of one ticket, and both revenues and prices are measured in dollars.[/size][br][br][size=150]Without creating a graph of [math]A[/math], determine which performance gives the greater maximum revenue when tickets are [math]p[/math] dollars each. Explain or show your reasoning.[/size]

IM Alg1.7.23 Practice: Using Quadratic Expressions in Vertex Form to Solve Problems

[size=150]Here is a graph of a quadratic function [math]f(x)[/math]. What is the minimum value of [math]f\left(x\right)[/math]?[/size][br][img]data:image/png;base64,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[/img][br]
[size=150]The graph that represents [math]f\left(x\right)=\left(x+1\right)^2-4[/math] has its vertex at [math]\left(-1,-4\right)[/math].[br][/size][br]Explain how we can tell from the expression [math]\left(x+1\right)^2-4[/math] that -4 is the minimum value of [math]f[/math] rather than the maximum value.
Each expression here defines a quadratic function. Find the vertex of the graph of the function. Then, state whether the vertex corresponds to the maximum or the minimum value of the function.
[math]\left(x-5\right)^2+6[/math]
[math]\left(x+5\right)^2-1[/math]
[math]\text{-}2(x+3)^2-10[/math]
[math]3\left(x-7\right)^2+11[/math]
[math]\text{-}(x-2)^2-2[/math]
[math]\left(x+1\right)^2[/math]
[size=150]Consider the equation [math]x^2=12x[/math].[br][br][size=100]Can we use the quadratic formula to solve this equation? Explain or show how you know.[/size][/size]
[size=100]Is it easier to solve this equation by completing the square or by rewriting it in factored form and using the zero product property? Explain or show your reasoning.[/size][br]
Match each equation to the number of solutions it has.
[math](x-1)(x-5)=5[/math]
[math]x^2-2x=\text{-}1[/math]
[math](x-5)^2=\text{-}25[/math]
[size=150]Which equation has irrational solutions?[/size]
[size=150]Let [math]I[/math] represent an irrational number and let [math]R[/math] represent a rational number. Decide if each statement is true or false. Explain your thinking.[br][/size][br][math]R\cdot I[/math] can be rational.
[math]I\cdot I[/math] [size=100]can be rational. [/size]
[math]R\cdot R[/math] [size=100]can be rational. [/size]
[size=150]Here are graphs of the equations [math]y=x^2[/math], [math]y=\left(x-3\right)^2[/math], and [math]y=\left(x-3\right)^2+7[/math].[/size][br][img]data:image/png;base64,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[/img][br]How do the 3 graphs compare?[br]
How does the -3 in [math]\left(x-3\right)^2[/math] affect the graph?[br]
How does the +7 in [math]\left(x-3\right)^2+7[/math] affect the graph?[br]
Three $5,000 loans have different annual interest rates. Loan A charges 10.5% annual interest, Loan B charges 15.75%, and Loan C charges 18.25%.
If we graph the amount owed as a function of years without payment, what would the three graphs look like? Describe or sketch your prediction.
Use technology to graph each function.
Based on your graphs, if no payments are made, about how many years will it take for the unpaid balance of each loan to triple?

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