[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.
[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]
[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]
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.
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?