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.
IM Alg1.7.24 Practice: Using Quadratic Equations to Model Situations and Solve Problems
[size=150]The function [math]h[/math] represents the height of an object [math]t[/math] seconds after it is launched into the air. The function is defined by [math]h\left(t\right)=-5t^2+20t+18[/math]. Height is measured in meters.[br][br]Answer each question without graphing. Explain or show your reasoning.[br][/size][br]After how many seconds does the object reach a height of 33 meters?
When does the object reach its maximum height?[br]
[size=100]What is the maximum height the object reaches? [br][br][/size]
The graphs that represent a linear function and a quadratic function are shown here.
[size=150]The quadratic function is defined by [math]2x^2-5x[/math].[/size][br][img]data:image/png;base64,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[/img][br][br]Find the coordinates of [math]R[/math] without using graphing technology. Show your reasoning.
[size=150]Diego finds his neighbor's baseball in his yard, about 10 feet away from a five-foot fence. He wants to return the ball to his neighbors, so he tosses the baseball in the direction of the fence.[br][br]Function [math]h[/math], defined by [math]h\left(x\right)=-0.078x^2+0.7x+5.5[/math], gives the height of the ball as a function of the horizontal distance away from Diego.[/size][br][br][size=150]Does the ball clear the fence? Explain or show your reasoning.[/size]
[size=150]Clare says, “I know that [math]\sqrt{3}[/math] is an irrational number, so its decimal never repeats or terminates. I also know that [math]\frac{2}{9}[/math] is a rational number, so its decimal repeats or terminates. But I don’t know how to add or multiply these decimals, so I am not sure if [math]\sqrt{3}+\frac{2}{9}[/math] and [math]\sqrt{3}\cdot\frac{2}{9}[/math] are rational or irrational."[br][br]Here is an argument that explains why [math]\sqrt{3}+\frac{2}{9}[/math] is irrational. Complete the missing parts of the argument.[/size][br][br]Let [math]x=\sqrt{3}+\frac{2}{9}[/math]. If [math]x[/math] were rational, then [math]x-\frac{2}{9}[/math] would also be rational because [math]\ldots[/math].
But [math]x-\frac{2}{9}[/math] is not rational because [math]\ldots[/math].
Since [math]x[/math] is not rational, it must be [math]\ldots[/math].
[size=100]Use the same type of argument to explain why [math]\sqrt{3}\cdot\frac{2}{9}[/math] is irrational.[/size]
The following expressions all define the same quadratic function.
[table][tr][td][math]x^2+2x-8[/math][/td][td][math](x+4)(x-2)[/math][/td][td][math](x+1)^2-9[/math][/td][/tr][/table][br][br]What is the [math]y[/math]-intercept of the graph of the function?
What are the [math]x[/math]-intercepts of the graph?
What is the vertex of the graph?[br]
Sketch a graph of the quadratic function without using technology. Make sure the x-intercepts, y-intercept, and vertex are plotted accurately.
[size=150]Here are two quadratic functions: [math]f\left(x\right)=\left(x+5\right)^2+\frac{1}{2}[/math] and [math]g\left(x\right)=\left(x+5\right)^2+1[/math].[/size][br][size=100][size=150]Andre says that both [math]f[/math] and [math]g[/math] have a minimum value, and that the minimum value of [math]f[/math] is less than that of [math]g[/math]. [/size][br][br]Do you agree? Explain your reasoning.[/size]
[size=150]Function [math]p[/math] is defined by the equation [math]p\left(x\right)=\left(x+10\right)^2-3[/math].[br][br]Function [math]q[/math] is represented by this graph.[br][img]data:image/png;base64,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[/img][br][/size][br][br]Which function has the smaller minimum? Explain your reasoning.[br]
Without using graphing technology, sketch a graph that represents each quadratic function.
[size=150]Make sure the [math]x[/math]-intercepts, [math]y[/math]-intercept, and vertex are plotted accurately.[/size]