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.
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.
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.
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?Emily is a college student who plans to spend a summer in Seattle.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.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.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.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.