Optimal Value and Step Pattern QUADTRATIC RELATIONS Optimal Value and Step Pattern

Opens up- When A > 0 Opens DOWN- When A < 0

Optimal Value The height of the highest or lowest point Always the last number That is the maximum value if the graph opens down That is the minimum value if the graph opens up. The OPTIMAL VALUE always corresponds to the y coordinate of the vertex. To find the value of the optimal value: A) Find the line of symmetry B) find the vertex, by substitution (This is the optimal value)

OPTIMAL VALUE y = ax2 + bx + c The standard form of a quadratic function is: y = ax2 + bx + c y x The parabola will open up when the a value is positive. OPENS UP- When A > 0 a < 0 a > 0 If the parabola opens up, the lowest point is called the vertex (minimum). Remind students that if ‘a’ = 0 you would not have a quadratic function. The parabola will open down when the a value is negative. Opens DOWN- When A < 0 If the parabola opens down, the vertex is the highest point (maximum).

GrAPHING QUADRATICS IN Standard form COMPLETE QUESTION 6 and 7! Find the maximum and minimum values

QUESTION 9 MINIMUM MAXIMUM (X-INTERCEPTS)

QUESTION 10

STEP PATTERNS http://www.youtube.com/watch?v=4gMaw64RDlc http://www.youtube.com/watch?v=JPorKyVh58Q The first differences show us the step pattern of the parabola. (I.e. in the case of y = x2 it would have a 1,3,5 step pattern) More importantly, all parabolas with ‘a’ values of 1 or (-1) will have 1,3,5 step patterns It also tells us the direction of opening If the second differences are (+) the parabola opens up If the second differences are (-) the parabola opens down OVER 1 (right or left) from the vertex point, UP 1² = 1 from the vertex point OVER 2 (right or left) from the vertex point, UP 2² = 4 from the vertex point OVER 3 (right or left) from the vertex point, UP 3² = 9 from the vertex point OVER 4 (right or left) from the vertex point, UP 4² = 16 from the vertex point

Question 11 (A) A ball is thrown into the air. The height of the ball after x seconds in the air is given by the quadratic equation h= -5x2 + 30x + 3, where h is the height in metres. Find the maximum height of the ball.

Question 11 (B) Alvin shoots a rocket into the air. The height of the rocket is h=-5x2+200x, where h is the height in metres. Find the maximum height of the rocket.

Question 11 (D) The cost, C, in dollars, to hire workers to build a new playground at a park can be modeled by C = 5x2 – 70x + 700, where x is the number of workers hired to do the work. How many workers should be hired to minimize the cost?

Question 11 (E) Jeff wants to build five identical rectangular pig pens, side by side, on his farm using 32m of fencing. The area that he will evaluate is given by the equation, A= -3w2 + 16w, where A is the total area in m2, and w is the width of the pig pen in m. Determine the dimensions (length and width) of the enclosure that would give his pigs the largest possible area. Calculate this area.

Question 11 (F) Studies have shown that 500 people attend a high school basketball game when the admission price is $2.00. In the championship game admission prices will increase. For every 20¢ increase 20 fewer people will attend. The revenue for the game will be R= -4x2 + 60x + 100, where R is the revenue in dollars, x is the number of tickets sold. a) What price will maximize the venue? b) What is the maximum revenue? Known:   500 tickets $2 cost Cost-20¢ increase results in #sold - 20 people Find: The price that will maximize the revenue The greatest revenue NOTE: revenue = (cost of ticket)(# tickets sold) = -4x2 + 60x + 100 = -4(30)2 + 60(30) + 100 =-1700 x= −(60) 2(−4)