Honors Precalculus: Do Now
Convert each quadratic function to standard form. What is the vertex? Is
it a Min or Max? (do these without a calculator)
1.) h(x) = 5x2 + 30x + 4
2.) h(x) = -x2 -3x + 3
3.) Determine the min or max of the following function. You may use the
“shortcut”.
t(x) = 5x2 – 7x + 2
Math History of the Day!
• Who created the quadratic formula?
• The precursor to what is known today as the quadratic formula, was derived
by an Islamic mathematician named Mohammed bin Musa Al-Khwarismi.
He derived the formula at about the same time as an Indian mathematician
named Baskhara did. This took place between 700 and 1100AD.
• This early version of the quadratic formula was carried to Europe in 1100
AD by a Jewish Mathematician / Astronomer from Barcelona named
Abraham bar Hiyya. As the Renaissance raged on in Europe, interest and
attention began to be focused on unique mathematical problems. Girolamo
Cardano began to compile the work on the quadratic equation in 1545.
• Cardano was one of the best algebraists of his time. He compiled the works
of Al-Khwarismi and Euclidian geometry and blended them into a form that
allowed for imaginary number. This inclusion also allowed for the existence
of complex numbers.
Working with just variables?
Solve for x.
ax2 +bx + c = 0
Example 1:
Modeling with Quadratic Functions
Most cars get their best gas mileage when traveling at a relatively
modest speed. The gas mileage of M for a certain new car is modeled
by the function below where s is the speed in Mi/hour and M is the
mileage in mi/gallon. What is the car’s best car mileage? What is its
speed?
1 2
M (s) = - s + 3s- 31
28
Example 2:
QUADRATIC FUNCTION MODELING
FIND THE DIMENSIONS OF A RECTANGULAR FIELD THAT CAN
BE ENCLOSED WITH 3000 FEET OF FENCE AND HAS THE
LARGEST POSSIBLE AREA. (no calculators).
Example 3: MAXIMIZING PROFIT
• A VENDOR CAN SELL 275 SOUVENIRS PER DAY AT A
PRICE OF $2 EACH. THE COST TO THE VENDOR IS
$1.50 PER SOUVENIR. EACH 10 CENT INCREASE,
DECREASES SALES BY 25 PER DAY. WHAT PRICE
SHOULD BE CHARGED TO MAXIMIZE PROFIT.
Example 4: Maximizing Revenue
A hockey team plays in an arena that has a seating capacity of 15,000
spectators. With the ticket price set at $14, average attendance at
recent games has been 9500. A market survey indicates that for each
dollar the ticket price is lowered, the average attendance increases by
1000.
A.) Find a function that models the revenue in terms of ticket price.
B.)Find the Price that maximizes revenue from ticket sales.
C.) What ticket price is so high that no-one attends and no revenue is
generated.
Homework #13!
Page 230-232
Section 3.1 #64, 66, 75, 77