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

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

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  1. OCF.01.4 - Finding Max/Min Values of Quadratic Functions MCR3U - Santowski

  2. (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.

  3. (B) Review - Max/Min Values and Forms of Quadratic Equations • 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

  4. (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

  5. (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?

  6. (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

  7. (E) Homework • Nelson text, p314 - 316 • Q1ac, 5ac, 6,7,8,12,15,16

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