Linear, Exponential and Quadratic Models

Key Concepts

• Draw the function that models the data set when the first differences are constant.
• Model the function that best suits the data set if the second differences are constant.
• Draw the function that models the data set if the ratios of consecutive y-values are the same.
• Compare the linear, quadratic, and exponential functions.

Vertex form of the quadratic function 

• The function f(x) = a(x−h)²+k, a≠0 is called the vertex form of a quadratic function 
• The vertex of the graph g is (h, k). 
• The graph of f(x) = a(x−h)²+k is a translation of the function f(x) = ax² that is translated h units horizontally and k units vertically. 

Standard form of the quadratic function 

• The standard form of a quadratic function is ax²+bx+c = 0, a≠0 
• The axis of symmetry of a standard form of quadratic function f(x) = ax²+bx+c is the line x=−b/2a. 
• The y-intercept of f(x) is c. 
• The x-coordinate of the graph of f(x) = ax²+bx+c is –b/2a. 
• The vertex of f(x) = ax²+bx+c is (–b/2a, f(–b/2a)). 

Modelling with quadratic functions 

• We can relate real-life situations using quadratic functions. 
• To find the height of an object, we can use the vertical motion model. 

Linear function 

A linear function best modules the data when the first differences are constant. 

The difference between consecutive y-values is called the first difference. 

Example: Here, the first differences are constant. 

Quadratic function 

The data set in which the second differences are constant is best modelled by the quadratic function. 

The difference between consecutive first differences is called second differences. 

Example: Here, the first differences in the data are not constant. But, the second differences are constant.

Exponential function 

The data set in which the ratios of consecutive y-values are constant is best modelled by an exponential function. 

Example: The ratios of y-values of the data are constant. 

Exercise

• When does the function h exceed the function f and function g?

• Determine whether a linear, quadratic, or exponential function is the best model for the data given:

• A savings account has a balance of $1. Savings Plan A will add $1,000 to an account each month, and Plan B will double the amount each month.

• Which plan is better in the short run? For how long? Explain.
• Which plan is better in the long run?

Concept Map  

What have we learned

When the independent variables change by a constant amount.

• Linear functions have constant first differences
• Quadratic functions have constant second differences
• Exponential functions have a constant ratio