Quadratic functions are in the form ax2 + bx + c, and have fascinated mathematicians for centuries. They have some fundamental properties that will explored in this unit. Probably the best known example of quadratic models are for projectiles. Anything that is thrown/launched and is subject to the laws of gravity will likely follow a quadratic path.
Key Concepts
In this unit you should learn to…
Understand the general expression of a quadratic function \(y=ax^{2}+bx+c\) and understand the significance of the coefficients.
Understand that quadratic functions model situations where there is a constant rate of change
Derive and use quadratic models
Essentials
Slides Gallery
Use these slides to review the material and key points covered in the videos.
1. About quadratic
This is a general introduction to quadratic functions
2. Key points
Here we cover the key points and features of quadratic models.
3.The y - intercept
A short look at understanding y - intercepts and how to find them.
4.Axis of symmetry and vertex
This covers the ways of finding the axis of symmetry and the coordinates of the vertex of a quadratic model.
5. Working with functions on your GDC
This video uses the context of quadratic functions to cover the key GDC skills needed for working with functions.
Summary
This section of the page can be used for quick review. The flashcards help you go over key points and the quiz lets you practice answering questions on this subtopic.
Flash Card
Review this condensed 'key point' flashcard to help you check and keep ideas fresh in your mind.
Test yourself
Self checking quiz
Practice your understanding on these quiz questions. Check your answers when you are done and read the hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.
For the first seven questions of the is queuiz you need to refer to the quadratic function y = x2 -2x - 15 and its graph, drawn below,
1
What is the exact value of y, when x = 2.5? (do not include spaces in your answer)
Type the function in to your GDC, use the table menu, set the table to go up in steps of 0.5 and read the answer from the table. OR, use the the 'value' or 'y solve' tool on your GDC.
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2
What is the exact value of x, when y = 128? (do not include spaces in your answer)
Type the function in to your GDC, use the table menu, set the table appropriately and read the answer from the table. OR, use the 'x solve' tool on your GDC.
3
Which of the following is the correct factorisation of the given quadratic?
Expand the brackets to be sure. SInce the coefficient of x is -2, this must come from +3, -5. SInce the value of 'c' is - 15, this corresponds.
4
Use your GDC to find the 'zeros' or 'roots' of this quadratic equation. What are the coordinates of the positive root? Answer in the form (x,y) with no spaces.
USe the 'Calc' or 'gsolve' menu on your calculator and choose 'zeros' or 'roots'.
5
What are the coordinates of the y-intercept of the function? Answer in the form (x,y) with no spaces.
The y - intercept occurs when x = 0. This leaves y = -15.
6
What is the equation of the axis of symmetry for this function? (no spaces)
The axis of symmetry is given by x = -b/2a, where y = ax2 + bx + c
7
Use your GDC, or otherwise, to work out the coordinates of the vertex of the function. (no spaces)
Ues the 'calc' or 'gsolve' menu on your calculator and choose 'min'. Alternatively, sonce you know the axis of symmetry is x = 1, work out the value of y when x = 1.
8
Which of the following functions is shown in the diagram below?
We can see that the function intersects th y axis at (0,-5) and so c = -5. This narrows it down to 2 functions. You can see the the function has zeros at x = -2.5 and 1. These values of x should give y = 0 when substituted.
9
The profit (P) in dollars made by a given enterprise is given by where x is the number units sold. What was the cost of their intial investment? How many units need to be sold before they break even. (this means they have covered their costs - answer should be a whole number)
Cost of initial investment dollars
Number of units sold to break even units
The initial investment is the money they spent before anything was sold. This is the y value when x = 0 , in other words, the y-intercept. The number of units for breakeven is the value of the first zero. Up until that point, the profit is negative. 5 is the fisrt whole number x - value that yields a profit of greater than or equal to zero.
10
The profit (P) in dollars made by a given enterprise is given by where x is the number units sold. What is the maximum profit that can be made with this model? How many units need to be sold for this maximum profit?
Maximum profit
Number of units
The maximum profit is the y coordinate of the maximum point on the curve. The number of units is the corresponding x coordinate. Use your GDC to find these values.
Exam Style Questions
The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.
Question 1
Video solution
Question 2
Video solution
Question 3
Video solution
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MY PROGRESS
How much of 2.5 Quadratic Models have you understood?
Feedback
Which of the following best describes your feedback?
Parabolas
where x is the number units sold. What was the cost of their intial investment? How many units need to be sold before they break even. (this means they have covered their costs - answer should be a whole number)
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