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- Today you are going to learn about "Quadratic relations".
- You might see the AOS term in questions it means Axis Of Symmetry.
- Example 2: y=x +2x+2
- y=ax +bx+c
- 2
- And that dashed line is called "Axis of symmetry".The axis of symmetry of a parabola is a vertical line that divides the parabola into two equal halves.
- A quadratic relation is an expression of the form y=ax^2+bx+c , a≠0 and this curved line is called "Parabola". Parabola is the graph of a quadratic relation.
- x y
- Let's say we have to zeros and one zeros is "a" and the other one is "b". Equation of the axis of symmetry: a+b/2
- y= a(x-r) (x-s)
- y=ax +bx+c
- 2
- As you see these 2 red points show where the parabola crosses the x-axis. Those points are called "Zeros, roots or x-intercepts".
- And the blue point is the y-intercept. Y-intercept is where the parabola crosses the y-axis. And that means when the x=0 we can find the y-intercept.
- Equation of the axis of symmetry is also the x value of the vertex. We show like "x= 3".
- Example: y= (x-4) (x+2)
- These yellow points are called "vertex". The vertex of a parabola is the point where the axis of symmetry and the parabola intersect each other.
- It is the point where the parabola is at it's maximum and minimum value.
- Example 1: y=x +2x+1
- Example 2: y=-x +2x+2
- 1) y=ax +bx+c
- 2
- 2
- 2
- First equation is the standard form of the quadratic relation as you see the greatest exponent 2 is exponent of x. The degree of the standard form is 2.
- Let's use the table of values to graph this 2 equations and see the differences.
- Example 1: y=x +2x+1
- 2
- Now let's find the 1st and 2nd differences.
- x
- -3 4
- -2 1
- -1 0
- 0 1
- 1 4
- y
- As you see when the second differences are the same but 0 relation is quadratic.
- Example 1: y=x +2x+1
- 2
- -3
- -1 2
- 3 2
- 1 2
- 1st 2nd
- When you look at the 2 parabolas what do you see?If you look at the directions of parabolas and coefficients of x^2s when the coefficient of x^2 positive (+) the parabola opens up and coefficient of x^2 negative (-) the parabola opens down.
- 2
- -1 0
- 0 2
- 2 0
- 0.5 2.25
- 1 2
- This is the factored form of the quadratic relation. "a" is the same with coefficient of x^2 "a" in the standard form as we have learned coefficient of x^2 defines the direction of the parabola.
- "r" and "s" represent the zeros of the parabola. Let's look at an example.
- y= a(x-r) (x-s)
- Equation of AOS: x=1 and it is also the x value of the vertex.Y- intercept: x=0, y=(-4) (2)= -8The vertex: x=1, y=(-3) (3)= -9(y value of he vertex), (1,-9)
- We can find the zeros by using the factored from.The zeros are 4 and -2. As we have learned we can find the AOS 4+ (-2)/2= 1.
- y= a(x-r) (x-s)
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