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) = 5x x ) h(x) = -x 2 -3x ) Determine the min or max of the following function. You may use the “shortcut”. t(x) = 5x 2 – 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.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 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 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. ax 2 +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?

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 A market survey indicates that for each dollar the ticket price is lowered, the average attendance increases by 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 Section 3.1 #64, 66, 75, 77