Solving Problems With Linear Functions

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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Let’s briefly review them: Identify changing quantities, and then define descriptive variables to represent those quantities.

When appropriate, sketch a picture or define a coordinate system.

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

We must be careful to analyze the information we are given, and use it appropriately to build a linear model.

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.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.In the above example, we were given a written description of the situation.We followed the steps of modeling a problem to analyze the information.Then we can substitute the intercept and slope provided.To find the x-intercept, we set the output to zero, and solve for the input.Some real-world problems provide the y-intercept, which is the constant or initial value.Once the y-intercept is known, the x-intercept can be calculated.Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.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.