1. Non-Linear Functions Senior 3 Applied Math Unit A
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©RAF, Nov 2005

2. Graph y = x2 Vertex y x y 3 9 2 4 1 1 x -1 1 -2 4 -3 9
The vertex of a parabola is where it reaches a maximum or minimum value 2 4 1 1
Vertex x -1 1 -2 4
The vertex is a point. In this case it is Point (0, 0) -3 9

3. y = x2- 6x + 5 Roots Vertex y x y 6 5 5 4 -3 x 3 -4 2 -3 1 5
Y-intercept x y
Line of Symmetry 6 5
The y-intercept is where the curve crosses the y-axis. 5
Roots
Or x-intercepts 4 -3 x 3 -4
The ‘roots’ or x-intercepts, are where the curve touches the x-axis
Vertex 2 -3 1
The Line of Symmetry is the line about which the curve is mirror image 5

4. Vertex When graphing parabolas the most important point is the vertex.
In Applied Math we use graphing tools (the TI-83) to find the vertex
We can also sometimes use a formula on the next slide to find the vertex

5. The vertex is located at the point
When the parabola is written with this style
y = f(x) = ax2 + bx + c
The vertex is located at the point

6. Vertex of y = ax2 + bx + c y = x2 + 8x + 15 y = (-4)2 + 8(-4) + 15
Find the x value of the vertex
‘Plug’ that value back into the original equation and find y.
a = 1
b = 8
c = 15
y = x2 + 8x + 15
y = (-4)2 + 8(-4) + 15
y = = -1

7. Grab a Pencil and Paper Identify the Vertex of these Quadratic Equations
y = x2 - 6x + 4
y = x2 + 10x + 25
y = x2 - 8x + 23
(3, -5)
(-5, 0)
(4, 7)
x = -(-6)/2= +3
y =(3)2-6(3)+4= -5
x = -(10)/2= -5
y =(-5)2-10(-5)+25= 0
x = -(-8)/2= +4
y =(4)2-8(4)+23=7

8. Direction of the Parabola
If the coefficient of x2 is positive the parabola will open up. 
If the coefficient of x2 is negative the parabola will open down. 
If the parabola opens up, it will have a minimum. If it opens down it will have a maximum

9. Which way does it Open? 3x2 - 4x + 2 -2x2 - 5x + 2 -(3/8)x2 + 2x - 4Up
Down

10. Finding y-intercept Use the TI-83 CALC function with a‘VALUE’ of ‘0’
There will always be one and only one y-intercept for a quadratic equation and its parabola
Use the TI-83 CALC function with a‘VALUE’ of ‘0’
Or ….just substitute in ‘0’ for x in the quadratic equation of the parabola
(Since the y-axis is where x = 0)
Example: find the y-intercept of
y = x2 + 2x + 3
y = (0)2 + 2(0) + 3 = 3

11. Practice Finding y-intercept from equation
There will always be one and only one y intercept for a quadratic equation and its parabola
Try these without the TI-83. Find the y-intercept of
y = 3x2 - 2x + 9
y = 3(0)2 – 2(0) + 9 = 9
y = *x x - 2
y = *(0) *(0) - 2 = -2

12. Roots or x-intercept of a quadratic
A quadratic equation and its parabola can have either 0, 1 , or 2 ‘roots’. A root is the same as an x-intercept, where the parabolic curve touches the x-axis, or in other words, where the y is equal to zero.
The x-axis is the same as the line y=0

13. Find x-intercept(s) of quadratic equations
The x-intercept is also called a root. A quadratic equation and its parabola can have either 0, 1, or 2 roots
In Grade 11 we only need to use the TI-83 to find the roots. Just use the CALC , ZERO function on the TI-83.

14. Finding Roots by factoring
You don’t need to know this ‘factoring to find roots’ in Grade 11 Applied Math. Skip if you want
But sometimes you can easily factor an equation, especially a quadratic, into two binomials. Example: x2 – x - 2 = (x-2)*(x+1)
So when does y, which is (x-2)*(x+1) = 0 ?
When x is 2 and when x is –1. Both will make the y equal to zero. So the roots are x = 2 and –1
If you want to get really advanced beyond Grade 11 Applied try looking up: ‘quadratic formula’ also

15. Domain and Range
Recall Domain and Range from Grade 10. It was in the Relations and Functions Unit
Domain is all the values the input values (x) can take on
In a quadratic equation (parabola curve) x can be all values from - to + . There is nothing to stop us making x whatever we want. We say the domain is: <x< 
Range is all the values the output (y) can take on
Notice that all quadratics have a maximum or a minimum. So the y may not take on some values ever

16. Range and Domain Practice
Vertex is at {-b/2a , f(-b/2a)}
= {-2, -7}
x can take on any value
But y will never be less than -7 but it will still go to infinity upward
Domain: - < x < 
Range: -7  y < 
You can always use a graphing calculator to find the vertex too!

17. Range and Domain Practice
Vertex is at {-b/2a , f(-b/2a)}
= {-4, +12}
x can take on any value
But y will never be more than but it will still go to negative infinity downward
Domain: - < x < 
Range: -  < y  12

18. #1 Graph: y = x2-4x Domain: - < x <  Range: -4  y < 
(2, -4) Up -4
x = 2
(0, 0)
(4, 0)
Vertex
Opens
Minimum Value of y
Line of Symmetry
y- intercept
1st x-intercept or ‘zero’
2nd x-intercept or ‘zero’
Domain: - < x < 
Range: -4  y < 

19. #2 Graph: y = -x2 - 6x Domain: - < x <  Range: - < y  9
(-3, 9)
x = -3
(0, 0)
(-6, 0)
Vertex
Line of Symmetry
Y- intercept
1st X-intercept
2nd X-intercept
Domain: - < x < 
Range: - < y  9

20. #3 Graph: y = -2x2 - 4x + 4 Domain: - < x < 
(-1, 8)
x = -1
(0, 6)
(-3, 0)
(1, 0)
Vertex
Line of Symmetry
Y- intercept
1st X-intercept
2nd X-intercept
Domain: - < x < 
Range: - < y  8

21. #4 Graph: y = x2 + 4x + 3 Domain: - < x < 
(-2, -1)
x = -2
(0, 3)
(-3, 0)
(-1, 0)
Vertex
Line of Symmetry
y- intercept
1st x-intercept
2nd x-intercept
Domain: - < x < 
Range: -2  y < 

22. #5 Graph: y = 2x + 8 - x2 Or y = - x2 + 2x + 8 in the standard form
(1, 9)
x = 1
(0, 8)
(-2, 0)
(4, 0)
Vertex
Line of Symmetry
Y- intercept
1st X-intercept
2nd X-intercept
Domain: - < x < 
Range: - < y  9

23. #6 Graph: y = 6 – 2x2 - 4x y = - 2x2 – 4x + 6 (-1, 8) Vertex x = -1
(0, 6)
(-3, 0)
(1, 0)
Vertex
Line of Symmetry
Y- intercept
1st X-intercept
2nd X-intercept

24. Application of Quadratic Equation
The most common application of the quadratic equation is for ‘ballistics’. How high will something go if you throw it straight up at a certain speed?
The equation on earth is ‘height (h) = vt - 5t2 ’
v is the initial speed you throw the ball upwards , usually about 40 meters per second unless you are a professional baseball player
t is the time in seconds from when you release the ball
h is the height above the ground of the ball in meters

25. Ballistics At what time does the ball reach the vertex??
How high does the ball go??
What are the roots or zeros? (that is: when is the ball touching the ground)?

26. Higher Degree Equations
A Quadratic Equation is said to have a degree of 2, since the highest exponent on a variable is a 2.
Do not confuse quad with ‘4’, like an ATV has wheels, quad means ‘square’ in this sense
The next highest order equation is a cubic equation. It is a polynomial with highest exponent of 3 on a variable.

27. Cubic Equations A cubic equation has the form y=ax3 + bx2 + cx + d
Notice a Cubic Equation has two bumps in it!
A quadratic, of degree 2, had only one ‘bump’!
A line, of degree 1, had zero ‘bumps’
A cubic equation has the form
y=ax3 + bx2 + cx + d
Some samples are below

28. Applications of Cubic Equations
Cubic equations often relate to volume and weight types of problems
The mass of a fish is related directly to the cube of its length for example
The amount of lumber in a tree is related to the cube of its height

29. Weight of a Largemouth Bass
The weight, W, of a large mouth bass has been found to be: W = L3 / 2700
Where W is the weight in lbs and L is the length in inches
It is only a model, it isn’t a perfect equation
The equation of the model was found by doing a cubic ‘regression’ of data and measurements
You already know how to do a ‘linear regression’ from Grade 10!
A cubic regression is just a different selection on the TI-83 graphing calculator
A cubic equation seems to fit this type of data best, but it is still not perfect

30. Weight of Largemouth Bass
A good regression model, a cubic regression
An old model of weight vs length

31. Exponential Functions
We know all about 1st degree (lines), 2nd degree (quadratics and parabolas), and 3rd degree (cubic) equations now.
They are just polynomials!
There is another type of relationship between variables; the exponential relationship
It tends to be recursive (just like in your grade 10 EXCEL studies)
The exponential relationship has the form
y = a(b)x

32. y = a(b)x
Notice now that the independent variable, x, is now an exponent!
This is not a polynomial!
Things that multiply (or divide) themselves by a certain percentage every certain amount of time are exponential
Your Savings Bonds, the world population, the mould in your tub, the effect of inflation
Nuclear waste radiation is an example of something that reduces itself by a certain percentage every 10,000 years or so

33. The Mouldy Tub!
Some moulds and viruses can double themselves every few minutes or hours.
Check out the pretend example below for mould
Time [hrs]
# of Mould Spores
100 1
200 2
400 3
800

34. Best Fit Equation and Regression
You can see that the dependent variable doubles itself every hour. In other words
Time
Nbr of Spores
Exponent on base 100
100
=100 1
200
=100*2 2
400
=100*2*2
=100*22 3
800
=100*2*2*2
=100*23
So the formula is 100*2t ,where t is time in hours

35. Exponential Regression
You will recall from Grade 10 how to do a regression on a line
The TI-83 allows you to do an Exponential Regression also, exactly the same except you just choose a regression type as : Exponential

36. The Mouldy Tub Equation
The TI-83 will tell you that the equation is:
y=a*b^x
a=100
b=2

37. Radioactive Waste Example
Some forms of radioactive waste decay to half their amount every 10,000 years
There is a plan to dispose of radioactive waste in the northern parts of Ontario and Manitoba
You can use the things you learned here to model a formula for how much radioactivity there is after a certain time

38. The Effect of a and b y = a*(b)x

39. The Effect of a and b in y=a*bx
Remember the Uncle Leo puzzle? Would you rather be paid $10.00 a day for three weeks or start with a penny and then double your pay each day.
It is definitely an exponential equation. Something doubles itself every day.
Now, to see the effect of the a and the b which would you rather have for a daily pay:
Start at $5 and double it each day for 21 days? Or;
Start at 1 penny and triple it each day for 21 days?

40. Solution – Uncle Leo’s a=$0.01 b=3 a=$5.00 b=2 a=$5.00 b=2 a=$0.01 b=3
It starts out the first few days that the $5 method was better
But after about 16 or 17 days, the b=3 (tripling) equation really takes over! Your last daily pay using the penny and tripling method ends up being $35 Million Dollars!

41. Summary – Quadratic Characteristics
Line of symmetry
If ‘a’ is positive, parabola opens ‘up’
y -intercept
roots (x-intercepts)
You can find all the points using a TI-83 Calculator too

42. Summary - Quadratics Math
Line of symmetry: x = 4
y-intercept ; where x = 0
y =(0)2 - 8(0) + 12 = 12
Roots: Use TI-83 to find roots or graph. 2 and 6 in this case are the roots

43. Summary - Types of Equations
Linear : y = ax + b
You remember ‘mx+b’
Quadratic: y = ax2 + bx + c
2nd degree
Cubic: y = ax3 + bx2 + cx + d
3rd degree
Exponential: y = abx
Things that change by a certain percentage of themselves especially with time

44. I hope this helped you in your study of Grade 11 Non-Linear Functions
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I hope this helped you in your study of Grade 11 Non-Linear Functions