Quadratics

Brandon.H MPM2DO-A

Some things come up and go down, they have x-intercepts, a vertex, axis of symmetry and an optimal value. In quadratics there are 3 forms being:vertex form-y=a(x-h)...squared...+k, standard from and factored form.

Vertex Form

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For the axis of symmetry x=h.K represents the optimal value.Types of transformations are shown below.Sub y=0 in order to find the x-intercepts or zeroes.Step pattern is the vertical stretch or compress for graphing the parabola.

Standard Form

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Use the quadratic formula to find the zeroes.For the AOS x=-b/2a.To find y/Optimal value, sub in x.Complete the square as shown below to turn to vertex form.

FActored Form

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The zeroes/x-intercepts are r and s.For AOS x=r+s/2.Sub in x-value for optimal value.
How to Graph Parabolas

Quadratics 1

Topics:

-Important parts of a parabola: x-int/roots/zeroes, axis of symm, optimal value, max/min, vertex, opening up/down

-Graphing vertex form
-Finding an equation in vertex form given a point and vertex.
-Transformations
-Word problems
-Factored form graphing

-Second differences

-Step pattern

Importants Part of a Parabola

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These are all of the parts in a parabola.There are problems where there are missing values in some cases.The 3 forms can be used to get each of these part in the parabola.

Second Differences

The Second differences determine whether or not the graph will be linear or quadratic.

The step pattern is the vertical stretch/compress in a parabola.The original step pattern is Over1 Up1, Over2 Up4.This changes the vertical according to the A value.

Graphing Vertex Form

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When graphing vertex form(y=a(x-h)2+k), you must insert the given values into the equation.The x-intercept is located in the bracket, but when graphing you must flip the sign.For example the x intercept in ...(x-2)2+5.The x-intercept is 2 and y intercept is 5.In the equation above the -3 is the vertical stretch(affects the step pattern).The original step pattern is over1 up2, but since there is a -3, you would go over1 down-3, then over2 down -6 and so on if need be.

Finding an equation in Vertex Form given a point and a vertex

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The equation above tells you the vertex is on the point (-2,-4) therefore the equation would be y=a(x+2)2 -4.It passes through (1,9) so you would sub them into the equation as shown above and then solve for A, and add the vertical stretch into the equation and apply it to the step pattern in the graph.

Transformations

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Word Problems

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Group the 2 similar values together.Money and Members are the 2 categories/brackets in the case above.Then common factor if possibly and set each brackets = to 0 in order to find the x-values.Then find the AOS by adding the 2 x-values together and dividing by 2.Sub x into the equation to find y/Optimal Value.

Factored form Graphing

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Set each bracket equal to 0 for finding the zeroes.Find AOS for x and sub in for y.Vertical stretch determines the position the parabola is facing in the case above it is upward.

Quadratics 2:

Topics:

-Multiplying Binomials

-Common Factoring

-Factoring by Grouping
-Factoring Simple Trinomials
-Factoring Complex Trinomials
-Factoring Special Trinomials- Difference of Squares, Perfect Squares.
-Applications- Shape problems (ex. Area), motion problems(when will... hit the ground)
-Graphing Standard form

-Solving By factoring

Multiplying Binomials

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Use distributive property from 1 bracket to the next in order to get a solution in standard form.

Common Factoring

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Factor out the highest possible number and exponent that can multiply to each term or number and give the starting problem.

Factoring by Grouping

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When the brackets have the same terms...(x+9)(x+9) and also have a common factor then you must use 1 bracket and the common factored term like shown above.

Factoring Simple Trinomials

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Use trial and error to List possible values that will add to BX and multiply to C. AX+BX+C

Factoring Complex Trinomials

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Use the Trial and error method although the value of A in AX+BX+C is greater than 1.

Difference of Squares

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Square each number once and in the first bracket - the 2nd number and in the second bracket + the second number.

Perfect Squares

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When you have 2 brackets with the same terms you must have only 1 bracket but squared as shown above.

Applications

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A=lw is the second equation and the first must be created using the total value. Then solve for the maximum area by using the quadratic equation in standard form as shown above.

Graphing Standard Form

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x-b/2a=x and sub in equation for y. When graphing plot the vertex then apply the step pattern also use the AOS to assist you and show your work.

Solving by Factoring

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Set each bracket equal to 0.The signs of the x values are flipped.If x is more than 1, you would divide the second term by the first and also flip the signs to get the x-value.

Quadratics 3:

Topics:

-Completing the square

-Solving from vertex form

-Quadratic Formula

-Discriminant

-Word Problems

Completing the Square

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In standard form bracket the first 2 terms (ax+bx)+c.Then divide BX by 2 and square the number.You cannot just add a number to an equation because it would change the entire equation, although you can + and - a the same number from and equation.This will assist you toward getting the trinomial.Now that you ave 4 numbers and terms in the bracket you must only have 3 so you would multiply the vertical stretch by the last number in the bracket which would bring it out of the bracket.Use trial and error to find the square then add or subtract the numbers outside the bracket to complete vertex form.Shown above.
3 15 Quadratic Formula

Solving from Vertex Form

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Plot the point of the vertex then apply the step pattern to the parabola.

Quadratic Formula

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In the equation -b+/-...square root b2-4ac. Sub in the values from standard form and begin to solve for the x-values.
3 15 Quadratic Formula

Discriminant

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Square root---b2-4ac/2.If you get a positive number, you have 2 real solutions.If u have a - number there are no solutions.If your number is 0, there is only 1 solution.

Word Problems

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Group the terms in brackets.Convert to Standard form.For the revenue you can complete the square if u chose this method and the vale of k is the maximum revenue.For cost, sub the x-value you got in vertex form in the bracket relating to cost and solve.Then graph using vertex form and the stretch.

How do all 3 Quadratic Units connect?

Standard,factored and vertex form connect because vertex form gives the location of the the vertex and the maximum value also the x-value where it can be used to solve a word problem,Standard form has various methods to find the x-intercepts and the vertex, and Factored form shows the x-intercepts and in some cases the vertical stretch.All of these forms can be used for problem solving when you have to find certain values.If a quadratic relation is in standard form and cannot be factored fully, you can use partial factoring to help you determine the axis of symmetry of the parabola. Then you can use the axis of symmetry to determine the coordinates of the vertex. These forms/strategies can each be used to find certain parts of a parabola or in a word problem and give you the final solution.