Welcome to Modeling and Functions! This course is at the level after Algebra 1 and starts where the Algebra Through Puzzles course left off.

If you see a graph with sliders underneath, that means the sliders can be moved around to help solve the problem. The animation below demonstrates how this works. (Depending on your settings, you may need to click on the animation to see it move.)

The graph below shows world record times at running a mile from 1900 to 1960, as tracked by the International Association of Athletics Federations. The \(x\)-axis shows years where \(0\) is 1900; the \(y\)-axis shows times in seconds. Based on the data shown in the graph, and assuming the model is linear, approximately what should the world record at the mile have been in 2000?

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The graph below shows three snapshots of a model rocket (rocket not to scale) being launched at an angle. The \(x\)-axis shows horizontal distance in meters and the \(y\)-axis shows vertical distance in meters. Approximately how far away would you expect the rocket to land from its starting point?

Assume the rocket will travel in a parabolic path (without wind or other considerations). Both the end and tip of each rocket should be part of the graph.

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The previous problem depicted a **function**, because it related two sets of numbers (distance and height) such that if you checked any valid distance, the rocket would be at one (and only one) specific height. The distance in this case is the **input** of the function and the height is the **output**.

Formally, we can say the *height* is a function of the *distance*.

Is the reverse true, that the *distance* is a function of the *height*?

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Functions are processes that take some input and produce some output, where every valid input produces **one** specific output.

If inputs are on the horizontal \(x\)-axis and outputs are on the vertical \(y\)-axis on the graphs below, how many of them are functions?

The *domain* of a function is the set of all valid inputs.

The below graph is of \( f(x) = \sqrt{x^2 - a} .\) For which value of \(a\) is the domain of \( f(x) \) in the regions \( x \leq -4 \) and \( x \geq 4 ?\)

(Assume we are using real numbers only.)

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Functions are quite powerful in that they let us prove *general* things that apply to all graphs, even if we are uncertain what those graphs are.

In the graph below, you can change a base function \( f(x) \) with a value of \(Z.\) Which function corresponds with the situation?

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