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

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OCF.01.4 - Finding Max/Min Values of Quadratic Functions. MCR3U - Santowski. (A) Review - Max/Min Values. Recall that a parabola has a maximum if the parabola opens downward, which can be identified from an equation if the value of a is negative.

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

MCR3U - Santowski

(A) Review - Max/Min Values
• Recall that a parabola has a maximum if the parabola opens downward, which can be identified from an equation if the value of a is negative.
• Recall that a parabola has a minimum value if the parabola opens upward, which can be identified from an equation if the value of a is positive.
• Recall the various ways of using an equation to determine the location of the vertex:
• (1) Vertex form: y = a(x - h)² + k

 the vertex at (h,k)

• (2) Intercept form: y = a(x - s)(x - t)

 the axis of symmetry is halfway between s and t

 when the x value for the is substituted into the equation, you can find the coordinates of the vertex

• (3) Standard form: y = ax² + bx + c

 axes of symmetry is at x = -b/(2a)

 when the x value for the is substituted into the equation, you can find the coordinates of the vertex

• (3) Standard Form: y = ax² + bx + c

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

(C) Examples of Algebraic Problems
• (i) Find the max (or min) value of y = -0.5x2 - 3x + 1
• (ii) Find the max (or min) point of y = 1/10x2 – 5x + ¼
• (iii) Find the vertex of y = 3x2 – 4x + 6
(D) Examples of Word Problems
• ex 1. A ball is thrown vertically upward from a balcony of an apartment building. The ball falls to the ground. Its height, h in meters above the ground after t seconds is given by the equation h = -5t2 + 15t + 45.
• (i) Determine the maximum height of the ball
• (ii) How long does the ball take to reach the maximum height?
• (iii) How high is the balcony?
• ex 2. Last year, talent show tickets are sold for \$11 each and 400 people attended. It has been determined that a ticket price rise of \$1 causes a decrease in attendance of 20 people. What ticket price would maximize revenue?
(D) Examples of Word Problems
• ex 3. If you plant 100 pear trees in an acre, then the annual revenue is \$90 per tree. If more trees are planted, they generate fewer pears per tree and the annual revenue per tree is decreased by \$0.70 for each additional tree planted. Additionally, it costs \$7.40 per tree per year for maintaining each tree. How many pear trees should be planted to maximize profit?
• (i) What is the equation for revenue?
• (ii) What is the equation for profit?
• (iii) find the max value for the profit equation
(E) Homework
• Nelson text, p314 - 316
• Q1ac, 5ac, 6,7,8,12,15,16