Back to Basics -

Hey Guys! for this  (morning/afternoon/evening) I will tackled first a review regarding some previous stuffs from grade 11 about Quadratic Functions because I believe that before moving on another more complex topics we should go back to basics.  And I would also include about our previous lesson from yesterday the Transformation of Functions. Before moving on another lesson we should make sure that we are right on track so bare with me guys :)  

What my mom and dad used to say is that math is not about memorizing stuffs, but better yet its more of practicing stuffs.

They also told me that in life you really have to fail and to fail to become successful, because believe it or not failure builds character and will served as your stepping stone in life.

Study Plan 101:   

Remember this from way back from your grade 11 years... 

What are functions? 

- A function in math is the process of showing a relationship between the input and output of a problem. The number you put in is directly related to the number you get out.

f(x)= y

This means that for every  x value, there is only one y value

The most basic quadratic equation that you'll see  is x^2.

*note: A mind refresher for remembering stuffs.One way to graph a quadratic equation is using table of values. So this is the example of the table of values of the quadratic equation x^2. 

 

x	y
-2 -1  0  1  2	4 1 0 1 4

You can pick any number in the x-axis, and then plug them in your calculator solve it mentally. I chose this numbers for you to see how a parabola opens. Ex: your original equation is x^2. 

Always draw a nicely smooth curving line passing  neatly through the plotted points. And don't forget the arrow in the parabola

As for me this served as a guide for us to graph much complex and cooler quadratic equations. so keep this in mind and heart guys :)

* Second, you should know how to identify standard form from vertex form 

Graphing Quadratic Equations

The graphical representation of quadratic equations are based on the graph of a parabola. A parabola is an equation of the form y = a x² + bx + c. The most general parabola, shown at the right, has the equation y = x².
The coefficent, a, before the x² term determines the direction and the size of the parabola. For values of a > 0, the parabola opens upward while for values of a < 0, the parabola opens downwards. The graph at the right also shows the relationship between the value of a and the graph of the parabola.

The vertex is the maximum point for parabolas with a < 0 or minimum point for parabolas with a > 0. For parabolas of the form y = ax², the vertex is (0,0). The vertex of a parabola can be shifted however, and this change is reflected in the standard equation for parabolas. Given a parabola y=ax²+bx+c, we can find the x-coordinate of the vertex of the parabola using the formula x=-b/2a. The standard equation has the form y = a(x - h)² + k. The parabola y = ax² is shifted h units to the right and k units upwards, resulting in a parabola with vertex (h,k).

The standard form of a parabola's equation is generally expressed:

• y = ax ² + bx + c
  • The role of 'a'
    • If a> 0, the parabola opens upwards
    • if a< 0, it opens downwards.
  • The axis of symmetry
  The axis of symmetry is the line x = -b/2a
  
  

The vertex form of a parabola's equation is generally expressed as : 
y= a(x-h)²+k 

• (h,k) is the vertex as you can see in the picture below
• If a is positive then the parabola opens upwards like a regular "U".
• If a is negative, then the graph opens downwards like an upside down "U".

If |a| < 1, the graph of the parabola widens. This just means that the "U" shape of parabola stretches out sideways .

If |a| > 1, the graph of the graph becomes narrower(The effect is the opposite of |a| < 1).

You can see these trends when you look at how the curve y = ax2 moves as "a" changes:

As you can see, as the leading coefficient goes from very negative to slightly negative to zero (not really a quadratic) to slightly positive to very positive, the parabola goes from skinny upside-down to fat upside-down to a straight line (called a "degenerate" parabola) to a fat right-side-up to a skinny right-side-up.   

There is a simple, if slightly "dumb", way to remember the difference between right-side-up parabolas and upside-down parabolas:

positive quadratic y = x2	negative quadratic y = –x2

Graph the following parabola:

y = (x + 3)² + 4

The vertex is the point (-3,4) 

Axis of Symmetry x= -3

direction opens up because a is positive (+)

min value = y = 4

y-int= ( 0 + 3)^2 +4 = 13

 y = –(x–1)² + 1?

The vertex is the point (1,1) 

Axis of Symmetry x= 1

direction opens down because a is negative (-)

min value = y = 1

y-int= ( 1-1)^2 +1= 2

Example: Plot f(x) = 2x² - 12x + 16

First, let's note down:

• a = 2,
• b = -12, and
• c = 16

Now, what do we know?

• a is positive, so it is an "upwards" graph ("U" shaped)
• a is 2, so it is a little "squashed" compared to the x² graph

Next, let's calculate h:

h = -b/2a = -(-12)/(2x2) = 3

And next we can calculate k (using h=3):

k = f(3) = 2(3)² - 12·3 + 16 = 18-36+16 = -2

So now we can plot the graph (with real understanding!):

We also know: the vertex is (3,-2), and the axis is x=3

Just like Transformations in Geometry, you can move and resize the graphs of functions

You can move it up or down by adding a constant to the y-value:

g(x) = x² + C

Note: if you want to move the line down, just use a negative value for C.

• C > 0 moves it up
• C < 0 moves it down

 

You can move it left or right by adding a constant to the x-value:

g(x) = (x+C)²

Adding C moves the function to the left (the negative direction).

Why? Well imagine you are going to inherit a fortune when your age=25. If you change that to(age+4) = 25 then you would get it when you are 21. Adding 4 made it happen earlier.

• C > 0 moves it left
• C < 0 moves it right

An easy way to remember what happens to the graph when you add a constant:

add to y: go high
add to x: go left

BUT you must add C wherever x appears in the function (you are substituting x+C for x).

Example: the function v(x) = x³ - x² + 4x

Move C spaces to the left: w(x) = (x+C)³ - (x+C)² + 4(x+C)

You can stretch or compress it in the y-direction by multiplying the whole function by a constant.

g(x) = 0.35(x²)

• C > 1 stretches it
• 0 < C < 1 compresses it

 

You can stretch or compress it in the x-direction by multiplying x (wherever it appears) by a constant.

g(x) = (2x)²

• C > 1 compresses it
• 0 < C < 1 stretches it

Note that (unlike for the y-direction), bigger values cause more compression.

 

You can flip it upside down by multiplying the whole function by -1:

g(x) = -(x²)

This is also called reflection about the x-axis (the axis where y=0)

You can combine a negative value with a scaling.

Example: multiplying by -2 will flip it upside down AND stretch it in the y-direction.

 

You can flip it left-right by multiplying the x-value by -1:

g(x) = (-x)²

It really does flip it left and right! But you can't see it, because x² is symmetrical about the y-axis. So here is another example using √(x):

g(x) = √(-x)

This is also called reflection about the y-axis (the axis where x=0)

Summary

y = f(x) + C	C > 0 moves it up C < 0 moves it down
y = f(x + C)	C > 0 moves it left C < 0 moves it right
y = C·f(x)	C > 1 stretches it in the y-direction 0 < C < 1 compresses it
y = f(Cx)	C > 1 compresses it in the x-direction 0 < C < 1 stretches it
y = -f(x)	Reflects it about x-axis
y = f(-x)	Reflects it about y-axis

Examples

Example: the function g(x) = 1/x

Move 2 spaces up: h(x) = 1/x + 2

Move 3 spaces down: h(x) = 1/x - 3

Move 4 spaces to the right: h(x) = 1/(x-4) (play with the graph)

Move 5 spaces to the left: h(x) = 1/(x+5)

Stretch it by 2 in the y-direction: h(x) = 2/x

Compress it by 3 in the x-direction: h(x) = 1/(3x)

Flip it upside down: h(x) = -1/x

 Case 1: If k > 1, then  g(x) is a vertical expansion of scale factor k ( Stretch )

 Case 2: If 0 < k < 1, then  g(x) is a vertical compression of scale factor k (Shrink)

k > 1	0<k < 1

(Vertical shift) 

Note that:

   If k > 0, then  f(x) moves up by k units,

and If k < 0, then f(x) moves down by k units.

k = 3	k = -3

       Note:

          If k > 0; The translation is to the LEFT,     

        and if k < 0; The translation is to the RIGHT 

k > 0	k < 0

Note also that 

Sources:

Google: https://www.google.ca/

Mathisfun.com

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