QUADTRATIC RELATIONS

Optimal Value and Step Pattern

OPENS UP- WHEN A > 0OPENS 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)

y = ax2 + bx + c

The parabola will open down when the a value is negative.Opens DOWN- When A < 0

The parabola will open up when the a value is positive.

OPENS UP- When A > 0

OPTIMAL VALUE

y

x

The standard form of a quadratic function is:

a > 0

a < 0

If the parabola opens up, the lowest point is called the vertex

(minimum).

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

HTTP://WWW.YOUTUBE.COM/WATCH?V=4GMAW64RDLCHTTP://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

STEP PATTERNS

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 costCost-20¢ increase results in #sold - 20 people

Find: 

The price that will maximize the revenueThe greatest revenue  

NOTE: revenue = (cost of ticket)(# tickets sold)

= -4x2 + 60x + 100

= -4(30)2 + 60(30) + 100

=-1700

x=−(60)2(−4 )