# Problem Solving With Linear Functions Key Output: \(M\), money remaining, in dollars Input: \(t\), time, in weeks So, the amount of money remaining depends on the number of weeks: \(M(t)\) We can also identify the initial value and the rate of change.Initial Value: She saved ,500, so ,500 is the initial value for M.For example, here is a problem: Maddie and Cindy are starting their very own babysitting business.

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We followed the steps of modeling a problem to analyze the information.

However, the information provided may not always be the same. Other times we might be provided with an output value.

Also, because the slope is negative, the linear function is decreasing.

This should make sense because she is spending money each week.

A negative input value could refer to a number of weeks before she saved \$3,500, but the scenario discussed poses the question once she saved \$3,500 because this is when her trip and subsequent spending starts.

It is also likely that this model is not valid after the x-intercept, unless Emily will use a credit card and goes into debt.Because this represents the input value when the output will be zero, we could say that Emily will have no money left after 8.75 weeks.When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. In this case, it doesn’t make sense to talk about input values less than zero.Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.The domain represents the set of input values, so the reasonable domain for this function is \(0t8.75\).In the above example, we were given a written description of the situation. The problem should list the Y- intercept, a starting amount of something and a slope, or a rate of change. You can tell that you need to create a linear equation by the information the problem gives you.When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function.Let’s briefly review them: Identify changing quantities, and then define descriptive variables to represent those quantities.

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