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.
![Vertex form of the quadratic function](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/8-1-1024x739.png)
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.
![Modelling with quadratic functions](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-4550.png)
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.
![Linear function table](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-4551.png)
![Linear function graph](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-4543.png)
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.
![Quadratic function table](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/7-3.png)
![Quadratic function graph](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-4544.png)
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.
![Exponential function table](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/6-4.png)
![Exponential function graph](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-4545.png)
Exercise
- When does the function h exceed the function f and function g?
![exercise 1](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/5-5.png)
- Determine whether a linear, quadratic, or exponential function is the best model for the data given:
![exercise 2](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/4-6.png)
- 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
![Concept Map](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-4547-1024x356.png)
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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