1 OCF 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 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 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: Recall the various ways of using an equation to determine 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) (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  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 (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 (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 = -0.5x 2 - 3x + 1 (i) Find the max (or min) value of y = -0.5x 2 - 3x + 1 (ii) Find the max (or min) point of y = 1/10x 2 – 5x + ¼ (ii) Find the max (or min) point of y = 1/10x 2 – 5x + ¼ (iii) Find the vertex of y = 3x 2 – 4x + 6 (iii) Find the vertex of y = 3x 2 – 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 = -5t t 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 = -5t t (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 height? (ii) How long does the ball take to reach the maximum height? (iii) How high is the balcony? (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? 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? 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? (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

7 (E) Homework Nelson text, p Nelson text, p Q1ac, 5ac, 6,7,8,12,15,16 Q1ac, 5ac, 6,7,8,12,15,16