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OCF.01.4 - Finding Max/Min OCF.01.4 - Finding Max/Min Values of Quadratic FunctionsValues of Quadratic Functions

MCR3U - SantowskiMCR3U - Santowski

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(A) Review - Max/Min Values(A) Review - Max/Min Values

Recall that a parabola has a maximum if the Recall that a parabola has a maximum if the parabola opens downward, which can be parabola opens downward, which can be identified from an equation if the value of identified from an equation if the value of aa is is negative.negative.

Recall that a parabola has a minimum value Recall that a parabola has a minimum value if the parabola opens upward, which can be if the parabola opens upward, which can be identified from an equation if the value of identified from an equation if the value of aa is is positive.positive.

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(B) Review - Max/Min Values and Forms of (B) Review - Max/Min Values and Forms of Quadratic EquationsQuadratic Equations

Recall the various ways of using an equation to determine Recall the various ways of using an equation to determine the location of the vertex:the location of the vertex:

(1) Vertex form: y = a(x - h)² + k (1) Vertex form: y = a(x - h)² + k the vertex at (h,k)the vertex at (h,k)

(2) Intercept form: y = a(x - s)(x - t) (2) Intercept form: y = a(x - s)(x - t) the axis of symmetry is halfway between s and t the axis of symmetry is halfway between s and t when the x value for the is substituted into the equation, when the x value for the is substituted into the equation,

you can find the coordinates of the vertexyou can find the coordinates of the vertex (3) Standard form: y = ax² + bx + c (3) Standard form: y = ax² + bx + c

axes of symmetry is at x = -b/(2a) axes of symmetry is at x = -b/(2a) when the x value for the is substituted into the equation, when the x value for the is substituted into the equation,

you can find the coordinates of the vertexyou can find the coordinates of the vertex (3) Standard Form: y = ax² + bx + c (3) Standard Form: y = ax² + bx + c

convert to vertex form using the method of competing convert to vertex form using the method of competing the squarethe square

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(C) Examples of Algebraic Problems(C) Examples of Algebraic Problems

(i) Find the max (or min) value of y = -0.5x(i) Find the max (or min) value of y = -0.5x22 - 3x + 1 - 3x + 1 (ii) Find the max (or min) point of y = 1/10x(ii) Find the max (or min) point of y = 1/10x22 – 5x + ¼ – 5x + ¼ (iii) Find the vertex of y = 3x(iii) Find the vertex of y = 3x22 – 4x + 6 – 4x + 6

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(D) Examples of Word Problems(D) Examples of Word Problems

ex 1. A ball is thrown vertically upward from a ex 1. A ball is thrown vertically upward from a balcony of an apartment building. The ball falls to the balcony of an apartment building. The ball falls to the ground. Its height, ground. Its height, hh in meters above the ground after in meters above the ground after tt seconds is given by the equation  seconds is given by the equation hh = -5 = -5tt22 + 15 + 15tt + 45. + 45.

(i) Determine the maximum height of the ball(i) Determine the maximum height of the ball (ii) How long does the ball take to reach the maximum (ii) How long does the ball take to reach the maximum

height?height? (iii) How high is the balcony?(iii) How high is the balcony?     ex 2. Last year, talent show tickets are sold for $11 ex 2. Last year, talent show tickets are sold for $11

each and 400 people attended. It has been each and 400 people attended. It has been determined that a ticket price rise of $1 causes a determined that a ticket price rise of $1 causes a decrease in attendance of 20 people. What ticket price decrease in attendance of 20 people. What ticket price would maximize revenue?would maximize revenue?

  

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(D) Examples of Word Problems(D) Examples of Word Problems

ex 3. If you plant 100 pear trees in an acre, then the ex 3. If you plant 100 pear trees in an acre, then the annual revenue is $90 per tree. If more trees are annual revenue is $90 per tree. If more trees are planted, they generate fewer pears per tree and the planted, they generate fewer pears per tree and the annual revenue per tree is decreased by $0.70 for annual revenue per tree is decreased by $0.70 for each additional tree planted. Additionally, it costs each additional tree planted. Additionally, it costs $7.40 per tree per year for maintaining each tree. $7.40 per tree per year for maintaining each tree. How many pear trees should be planted to maximize How many pear trees should be planted to maximize profit?profit?

(i) What is the equation for revenue?(i) What is the equation for revenue? (ii) What is the equation for profit?(ii) What is the equation for profit? (iii) find the max value for the profit equation(iii) find the max value for the profit equation

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(E) Homework(E) Homework

Nelson text, p314 - 316 Nelson text, p314 - 316 Q1ac, 5ac, 6,7,8,12,15,16Q1ac, 5ac, 6,7,8,12,15,16