## 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) = ax^{2}that is translated h units horizontally and k units vertically.

### Standard form of the quadratic function

- The standard form of a quadratic function is ax
^{2}+bx+c = 0, a≠0 - The axis of symmetry of a standard form of quadratic function f(x) = ax
^{2}+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) = ax
^{2}+bx+c is –b/2a. - The vertex of f(x) = ax
^{2}+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

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