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
The standard form of a quadratic function is:
y = ax2 + bx + c
y
The parabola will open up
when the a value is
positive.
OPENS UP- When A > 0
a>0
If the parabola
opens up, the
lowest point is
called the vertex
(minimum).
x
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)
.
a<0
GRAPHING QUADRATICS IN STANDARD FORM
COMPLETE QUESTION 6 and 7!
Find the maximum and minimum values
QUESTION 9
MINIMUM
(X-INTERCEPTS)
MAXIMUM
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)