Problem Solving With Linear Functions Key

Identify a solution pathway from the provided information to what we are trying to find. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem. Clearly convey your result using appropriate units, and answer in full sentences when necessary. In her situation, there are two changing quantities: time and money.How can they calculate how much they will charge for an evening of babysitting?

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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.

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.

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.She has saved ,500 for her trip and anticipates spending 0 each week on rent, food, and activities.How can we write a linear model to represent her situation?When appropriate, sketch a picture or define a coordinate system.Carefully read the problem to identify important information.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 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.To find the x-intercept, we set the output to zero, and solve for the input.\[\begin 0&=−400t 3500 \ t&=\dfrac \ &=8.75 \end\] The x-intercept is 8.75 weeks.

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