IM Alg1.7.24 Lesson: Using Quadratic Equations to Model Situations and Solve Problems

Write an equation representing the line that passes through each pair of points.
[math]\left(3,3\right)[/math] and [math]\left(5,5\right)[/math]
[math]\left(0,4\right)[/math] and [math]\left(-4,0\right)[/math]
Solve this equation: [math]x+1=\left(x-2\right)^2-3[/math]. Show your reasoning.[br]
[size=150]The function [math]h[/math], defined by [math]h\left(t\right)=-5t^2+10t+7.5[/math], models the height of a diver above the water (in meters), [math]t[/math] seconds after the diver leaves the board. For each question, explain how you know.[br][/size][br][size=100]How high above the water is the diving board?[/size]
[size=100]When does the diver hit the water?[/size][br]
[size=100]At what point during her descent toward the water is the diver at the same height as the diving board?[/size][br]
[size=100]When does the diver reach the maximum height of the dive?[/size]
[size=100]What is the maximum height the diver reaches during the dive?[/size][br]
[size=150]Another diver jumps off a platform, rather than a springboard. The platform is also 7.5 meters above the water, but this diver hits the water after about 1.5 seconds.[/size][br][br]Write an equation that would approximately model her height over the water, [math]h[/math], in meters, [math]t[/math] seconds after she has left the platform. Include the term [math]-5t^2[/math], which accounts for the effect of gravity.
[size=150]Here are graphs of a linear function and a quadratic function. The quadratic function is defined by the expression [math]\left(x-4\right)^2-5[/math].[/size][br][img]data:image/png;base64,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[/img][br][br][br]Find the coordinates of [math]P,Q[/math], and [math]R[/math] without using graphing technology. Show your reasoning.
Close

Information: IM Alg1.7.24 Lesson: Using Quadratic Equations to Model Situations and Solve Problems