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Axis Of Symmetry (K=h)

 

What is the Axis of symmetry? The axis of symmetry of a parabola is a vertical line that divides the parabola into two corresponding halves. The axis of symmetry always passes through the vertex of the parabola. It is the X coordinate of the vertex. X=h.

Optimal Value (y=k)

The optimal value is the highest or the lowest point on the parabola. The optimal value, also known as the vertex of the parabola would be maximum if only the parabola opens down words. One opening upwards has a minimum value. It "Y" coordinate of the vertex.

Step Pattern

^For this equation the vertex of the Parabola would be (1,1). therfore the axis of symmetry is the line x=1

Example;

y=k

Ex;

y=a(x-h)²+k

y=-3(x+2)+27

y=27

Therefore, the optimal value for the equation

y=-3(x+2)²+27 is y=27

The step pattern determines which points a parabola crosses. This is pattern followed by a number squared, starting at 1. For example, an equation with its "a" being 2 will go over 1, up 2, then over 2, up 8. This was found using "a" multiplied by the step pattern of one.

Over 1, Up 1

Over 2, Up 4

Over 3, Up 9

Over 4, Up 16 

Both the "K" and "H" represent the vertx of the porabala

The "H" in this quadratic equation represents a Horizontal translation. meaning how far left or how far right the graph is shifted from x=0 also known as the orgin. if the Value of 'H" is Positive then the number would be negative. If "H" is a negative number, therefore the number will be transformed into positive. Therefore a switch occurs for the signs. 

Example: if the equation is y = 2(x - 4)^2 + 8, the value of h is 4, and k is 8.

 

The "K" in this equation represents the vertical transaltion, so how far up or down the graph is shifted from x=0 Or the orgin. In the case of figuring out where 'K" goes, you would just have to know that if the value is negative it stay negative and if the number is positive the number will be positve. The swtich of the signs DO NOT occur to figuring out the "K" value.

 

The "a" in this equation indicates the stretch/compression (The strtch or compression will multiply the verticle part of the step pattern. I t also represents whether the quadratic opens up or opens down. you could tell if it opens down or up by; A positive "a" draws a smiley, and a negative "a" draws a frowny. Or even you could think of just by your knowledge if the "a" value is positve then it would open up. if The value of "a" is negitive then it would go down.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

X-intercepts or Zeroes

To figure out what the "x" intercept is, you would have to subsitute "y" into zero. y=0.

Example; y=2(x-4)^2+8 this equation will then change into

                 0=2(x-4)^2+8 (sub "0" in for the variable "y" )

 

The x-intercepts of zeroes is where the lines of a parabola crosses the x-intercept. This can be determined by subbing y=0 and solving the equation.

Example :

 

 

Ex: y=(x-2)^2-9

0=(x-2)^2-9

9=(x-2)^2

+/-√9=x-2 

3=x-2  or -3=x-2

3+2=x      -3+2=x 

5,-1=x

 

 

 

 

What do all these variables

                   mean?

 

y = a(x - h)^2 + k

Transformations

VERTEX FORM

Heres a video that shows how graph using the step pattern

GRAPHING FROM  VERTEX  FORM

How can you graph this equation?

Y=a(x-h)^2+k

Use the table of Values to graph your points

Lets do an Example

Use the STEP PATTERN to plot the points or use the information you have from your graph

Square root both sides and since the left side can be divided equally with two positives and two negatives, these can be used to determine to x-intercepts

How does this connect to graphing?

This connects to graphing because it gives an understanding to graphing parabolas in not only one, but in three ways, with more accurate calculations.

Here is a video showing how to find the

zeroes or x- intercepts

Word Problem Examples

y=a(x-h)²+k

Mike throws a ball 1m above the ground. After 2 seconds, the ball reaches its maximum height of 5m.

a) Write an equation in vertex form for the quadratic function expressing the height and time of the ball.

y=a(x-h)²+k

1=a(x-2)²+5

1=a(0-2)²+5

1=a(-2)²+5

1=a(4)+5

1-5=a(4)

-4=a(4)

 4

-1=a

b) Find the y- and x-intercepts of this equation.

0=-(x-2)²+5

-5=-(x-2)²

-1

±√5=√(x-2)²

2.2=x-2     or

2.2+2=x

4.2=x

Therefore, the x-intercepts for equation y=-(x-2)²+5 is x=4.2 and x=-0.2.

y=-(0-2)²+5

y=-(4)+5

y=1

Therefore, the y-intercept of this equation is y=1.

 

 

 

 

 

 

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