Quadratics

Quadratic Relationships

Vertex Form y=a(x-h)²+k

A quadratic equation in vertex form is written y=a(x-h)²+k. When you graph a quadratic equation in vertex form, you have to write the vertex which is given in the equation, then look at the step pattern shown by the value of a. In this equation the vertex is represented by h and k. The h being the x value and the k being the y value. 'h' may be written as negative or positive, the sign would be reversed when writing it in the vertex. Occasionally you may be given the vertex, and a point, which means that you would have to solve for the 'a' value to complete the equation and graph.


  • vertex is (h,k) (The maximum or minimum point on the graph. It is the point where the graph changes direction)
  • y=k is the optimal value

  • x=h is the axis of symmetry (divides parabola in half)


The value of (a)--> determines the orientation and shape of the parabola

Orientation

If a>0, the parabola opens up

If a<0, the parabola opens down

Shape

If -1<a<1, the parabola is vertically compressed

If a>1 or a<-1, the parabola is vertically stretched


The value of (k) --> determines the vertical position of the parabola

If k>0, the vertex moves up by k units

If k<0, the vertex moves down by k units


The value of (h) --> determines the horizontal position of the parabola

If h>0, the vertex moves to the right h units

If h<0, the vertex moves to the left h units


Example:
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Factored Form y=a(x-r)(x-s)

A quadratic equation in factored form is written as y=a(x-r)(x-s). In this equation, to figure out the x-intercepts you put each factor = 0, then solve for x. Another way that you can figure out the x-intercepts is by reversing the signs for both 'r' and 's'. If 'r' and 's' are positive the x-intercepts would be negative and if they are negative then the x-intercepts would be positive. After you have figured out both of the x-intercepts, add both the x values and divide them by 2 (r+s/2). This will lead you to knowing x=h which is also known as the axis of symmetry. Once you have done this, you have to find out the 'k' which is the optimal value. To find out the optimal value you substitute the 'h' (axis of symmetry) value into the equation, the value of y is k.


In general if y=a(x-r)(x-s)

The zeros are found by setting each "factor" equal to zero so:

x-r=0 means x=r and x-5=0 means x=5


The axis of symmetry

- is the midpoint of the two zeros so x=r+s/2


The optimal Value

-is found by subbing the axis of symmetry value into the equation

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Common Factoring

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Simple Trinomial

Complex Trinomial

Perfect Squares & Difference of Squares

https://www.youtube.com/watch?v=vchb32724S8&list=UUHniwmd-AJhoX9dlrxZJ-QA

Quadratic Formula

The quadratic formula is used to solve for the x-intercepts. This requires this formula:


Example:

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Word Problems

1. A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway?
2. You run a canoe-rental business on the Ottawa River. You currently charge $12 per canoe and average 36 rentals a day. An industry journal says that, for every fifty-cent increase in rental price, the average business can expect to lose two rentals a day. What should you charge to maximize your income?

Unit Connections

Equations and Graphing

All three equations can be graphed


Completing the Square and Vertex From:

When completing the square you take a equation from standard from to vertex form. By Completing the square you end up with a quadratic equation from standard form to vertex form and this is the main reason why they both connect.

Reflection

I feel that i could have done better in this unit. I feel that i was lacking some knowledge in many parts. The quadratics unit was divided into smaller parts making it easier for us to understand and learn. I feel that the first unit to me was the easiest. As we moved into factoring i started struggling more. There were many methods of factoring which made me a bit confused. So i struggled during test/quizzes because i was confused on what type of method to use. Overall, i feel that i could have succeeded in this unit, by asking questions, getting involved in class discussions and coming in for extra help.