# Problem Solving With Linear Functions Key

What would be the x-intercept, and what can she learn from it?To answer these and related questions, we can create a model using a linear function.

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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.

Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships.

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

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. 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.For example, here is a problem: Maddie and Cindy are starting their very own babysitting business.They charge parents dollars right when they come in and for every hour they need to babysit a child.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.Rate of Change: She anticipates spending 0 each week, so –0 per week is the rate of change, or slope.Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable.Also, because the slope is negative, the linear function is decreasing.This should make sense because she is spending money each week.

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