1.3 Minimum or Maximum MCR 3U

Schedule 1. Attendance Minimum or Maximum Note 3. Practice A. Homework Questions (individual)

New Presentation Format  Please hold on to any feedback (positive or constructive) until the end of the lesson/presentation  You need to write down everything written in blue  Any comments written down in the text box can be written down if you wish (optional)  If you are ever away, ask me for the skeleton note but ask your peers for the complete notes taken in class  Do NOT wait until before a test or assessment to get any missing notes

1.3: Maximum or Minimum of Quadratic Function  Recall: Quadratic functions can be written in a number of different ways  Standard Form: y=ax 2 +bx+c  a=vertical stretch/compression (in all forms)  c=y-intercept  Factored Form of a Quadratic Equation: y=a(x-r)∙(x-s)  r and s=x-intercepts  Vertex Form: y=a(x-h) 2 +k  (h, k)=coordinates of the vertex  If a is positive k=minimum value, if a is negative then k=maximum value

1.Converting from one form to another: A. Factored/Vertex Form Standard Form I. Eg. Convert y=2(x-1)(x-2) (factored form) to standard form:  y=2[(x-1)(x-2)]  y=2(x 2 -3x+2)  y=2x 2 -6x+4 Expand -Expand the binomial (x-1)(x-2) -Distribute the coefficient 2 into the new trinomial (x 2 -3x+2)

1.Converting from one form to another: II. Eg. Convert y=(x+2) 2 +3 (vertex form) to standard form:  y=(x+2) 2 +3  y=x 2 +4x+4+3  y= x 2 +4x+7 -Expand the binomial (x+2) 2  (x+2)(x+2) -Collect like terms 4+3 -Simplify 4+3=7

1.Converting from one form to another: B. Standard Form Factored Form I. Eg. Convert y=x 2 -3x+2 (standard form) to factored form  y=x 2 -3x+2  y=(x )(x )  y=(x-1)(x-2) -Factor the trinomial ( x 2 -3x+2 ) -The first terms of my resulting binomials are x and x since the only factors of 1 are: 1 and 1 -Two numbers that add to give me (-3) and multiply to give me (2) are…(-1) and (-2) Factor

1.Converting from one form to another: C. Standard Form Vertex Form I. Eg. Convert y=x 2 +5x+7 to vertex form:  Divide the middle term of the trinomial (5) by 2 and then square it: y=x 2 +5x+(5/2) 2 +7  Add or subtract this amount (add if it is negative, subtract if positive): y=x 2 +5x+(5/2) 2 -(5/2) 2 +7  Factor the first three terms (it should be a perfect square trinomial): y=(x+5/2) 2 [-(5/2) 2 +7]  Simplify the remaining expression by adding the remaining terms: y=(x+2.5) Complete the square

1.Converting from one form to another:  Vertex Form: y=(x+2.5)  The vertex is (-2.5, 0.75)  The a value (1) is positive so the graph opens up and the vertex is a minimum

1.Converting from one form to another: II. Eg. Convert y=2x 2 +12x+7 to vertex form :  Factor out the coefficient of x 2 from the first two terms (2x 2 +12x): y=2(x 2 +6x)+7  Divide the middle term of the trinomial (6) by 2 and then square it: y=2(x 2 +6x+(6/2) 2 )+7  Add or subtract this amount (add if it is negative, subtract if positive): y=2(x 2 +3x+9)-(9)+7  Factor the first three terms (x 2 +3x+9) (it should be a perfect square trinomial): y=2[(x+3) 2 -9]+7

1.Converting from one form to another  Multiply the factored out coefficient by the amount being taken out (-9): y=2(x+3)  Simplify the remaining expression by adding the remaining terms (-18+7): y=2(x+3)  The vertex of this function is (-3, -11)  The a value (2) is positive so the graph opens up and the vertex is a minimum

Schedule 1. Attendance 2. Questions from HW  Creating an equation/completing the square 3. Finish 1.3 Max and Min  Practice

Announcement  Mini Test on Wednesday:  Topics:  Solving quadratic equations  Creating quadratic equations

2.Use Partial Factoring to Find the Vertex of a Quadratic Function A. Find the vertex of this function: y=-2x 2 +8x-3 I. Use only the first part of the function (y=-2x 2 +8x) since the x- coordinate of the vertex of both of the functions will be the same II. Factor the function  y=-2x(x-4) III. Substitute 0 for y, to solve for the x-intercepts  0=-2x(x-4) IV. The zero product property (If ab=0, then either a=0 or b=0) tells us that…  x=0 or 4, these are my roots/x-intercepts/solution

2.Use Partial Factoring to Find the Vertex of a Quadratic Function

3.Solve a Problem Involving a Minimum or Maximum  P. 31 #5 A. An electronics store sells an average of 60 entertainment systems per month at an average of $800 more than the cost price. For every $20 increase in the selling price, the store sells one fewer system. What amount over the cost price will maximize revenue?  Let x represent the number of increases in selling price  Revenue (R) = Quantity (Q) x Price (P)  Q=60-1x  P=800+20x

3.Solve a Problem Involving a Minimum or Maximum  R=(-x+60)(20x+800)  R= -20x x+1200x  R= -20x x  Maximum occurs at the vertex  Solve for the vertex by completing the square

Solution  R= -20(x 2 -20x )  R=-20(x-10)  R=-20(x-10)  The vertex occurs at (10, 50000), this means that the maximum revenue of $ occurs when there have been 10 increases in selling price.  P=800+20(10)  P=1000  This occurs when the price is $1000 over the cost.

Class Work p. 31 #3 *If you have any questions from the previous days homework come up to my desk and ask for clarification*