When needed, write a formula for the function. Solve or evaluate the function using the formula. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically. 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. We can use this information to define our variables, including units.
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. Then we can substitute the intercept and slope provided.
Because this represents the input value when the output will be zero, we could say that Emily will have no money left after 8. 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.
Here the domain refers to the number of weeks. 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. 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. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. 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. Using a Given Intercept to Build a Model 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. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents.
We can then use the slope-intercept form and the given information to develop a linear model. The x-intercept is 4 months, so it will take Hannah four months to pay off her loan.
Using a Given Input and Output to Build a Model Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output. Given a word problem that includes two pairs of input and output values, use the linear function to solve a problem. Identify the input and output values.
Convert the data to two coordinate pairs. Find the slope. Use the model to make a prediction by evaluating the function at a given x-value. Use the model to identify an x-value that results in a given y-value. Answer the question posed. This is one of the trickier problems in the function unit.
Watch carefully where we substitute the given number 4. Let's take a look. Example 1: Solving for x in a linear function Pretty easy, right?
This is really just a review of concepts that you've already learned. Once you figure out that you substitute 4 for f x , you solve this as a regular two step equation.
Anything that involves a constant rate of change can be nicely represented with a line with the slope. Indeed, so long as you have just two points, if you know the function is linear, you can graph it and begin asking questions! Linear Mathematical Models Linear mathematical models describe real world applications with lines.
Learning Objectives Apply linear mathematical models to real world problems Key Takeaways Key Points A mathematical model describes a system using mathematical concepts and language. Linear mathematical models can be described with lines. The equation and graph can be used to make predictions. Real world applications can also be modeled with multiple lines such as if two trains travel toward each other. The point where the two lines intersect is the point where the trains meet. Key Terms mathematical model: An abstract mathematical representation of a process, device, or concept; it uses a number of variables to represent inputs, outputs, internal states, and sets of equations and inequalities to describe their interaction.
Mathematical Models A mathematical model is a description of a system using mathematical concepts and language. Linear modeling can include population change, telephone call charges, the cost of renting a bike, weight management, or fundraising.
After the model is written and a graph of the line is made, either one can be used to make predictions about behaviors. Real Life Linear Model Many everyday activities require the use of mathematical models, perhaps unconsciously. One difficulty with mathematical models lies in translating the real world application into an accurate mathematical representation. How much would a 75 mile trip cost? At what time will the two trains meet? At this time how far did the trains travel? To find where this is, plug [latex]2.
Plugging it into the first equation gives us [latex]50 2. The two trains meet at the intersections point [latex] 2.
Linear regression attempts to graph a line that best fits the data.
To determine the rate of change, we will use the change in output per change in input. While we could use the actual year value as the input quantity, doing so tends to lead to very cumbersome equations because the y-intercept would correspond to the year 0, more than years ago! However, the information provided may not always be the same. How long after they start walking will they fall out of radio contact? This allows assumptions about how the data is roughly spread out and predictions about future data points.
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. Identify the input and output of each linear model. Indeed, so long as you have just two points, if you know the function is linear, you can graph it and begin asking questions! Should I draw diagrams when given information based on a geometric shape?