Solving Problems With Linear Functions

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Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. She plans to pay $250 per month until her balance is $0.

The y-intercept is the initial amount of her debt, or $1,000. We can then use the slope-intercept form and the given information to develop a linear model.

\[\begin 0&=−400t 3500 \\ t&=\dfrac \\ &=8.75 \end\] The x-intercept is 8.75 weeks.

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.

In this section, we will explore examples of linear function models.

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.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.The amount of money she has remaining while on vacation depends on how long she stays.This should make sense because she is spending money each week.The rate of change is constant, so we can start with the linear model \(M(t)=mt b\).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 ,500, but the scenario discussed poses the question once she saved ,500 because this is when her trip and subsequent spending starts.To answer these and related questions, we can create a model using a linear function.Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships.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.

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