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4: Linear and Polynomial Functions

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    Recall that a function is a relation that assigns to every element in the domain exactly one element in the range. Linear functions are a specific type of function that can be used to model many real-world applications, such as plant growth over time. In this chapter, we will explore linear functions, their graphs, and how to relate them to data.

    • 4.1: Introduction to Linear Functions
      Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour. A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function.
    • 4.2: Linear Functions
      The ordered pairs given by a linear function represent points on a line. Linear functions can be represented in words, function notation, tabular form, and graphical form. The rate of change of a linear function is also known as the slope. An equation in the slope-intercept form of a line includes the slope and the initial value of the function. The initial value, or y-intercept, is the output value when the input of a linear function is zero.
    • 4.3: Modeling with Linear Functions
      We can use the same problem strategies that we would use for any type of function. When modeling and solving a problem, identify the variables and look for key values, including the slope and y-intercept. Draw a diagram, where appropriate. Check for reasonableness of the answer. Linear models may be built by identifying or calculating the slope and using the y-intercept. The x-intercept may be found by setting y=0, which is setting the expression mx+b equal to 0.
    • 4.4: Fitting Linear Models to Data
      Scatter plots show the relationship between two sets of data. Scatter plots may represent linear or non-linear models. The line of best fit may be estimated or calculated, using a calculator or statistical software. Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. The correlation coefficient, r , indicates the degree of linear relationship between data.
    • 4.5: Quadratic Functions
      In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.
    • 4.6: Power Functions and Polynomial Functions
      Suppose a certain species of bird thrives on a small island. The population can be estimated using a polynomial function. We can use this model to estimate changes in the bird population. In particular, we could use this model to predict when the bird population will disappear from the island. In this section, we will examine functions and explore using them to estimate and predict changes.
    • 4.7: Graphs of Polynomial Functions
      Suppose the revenue for a fictional cable company can be modeled by a polynomial function. We can study this function (and its graph) to answer questions regarding the company's revenue. Many things in life can be modeled using polynomials. We have already explored quadratic functions, a special case of polynomials. In this section, we  explore the behavior of polynomials in general.

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