Lesson 2-4: Quadratic Functions Day #2

1. Lesson 2-4: Quadratic Functions Day #2 Advanced Math Topics Mrs. Mongold

2. Quadratics and Word Problems • Real world applications for quadratics: • We are going to use quadratic functions to write equations for real world problems. • We are going to use this information to answer questions. • We are going to use our calculators to make life easier

3. Think about it… • What real world situations can we apply to parabolas? What types of things could you do with an object that would form a parabola? • Throw a ball • Drop something from a certain height

4. Some species of gulls use a clever technique for eating clams and other shellfish. In order to break a clam’s shell a gull will pick up the clam with its beak, fly into the air, and drop the clam onto a rock. If the shell does not break, the gull will drop the clam from greater heights until it is successful. Example 1 t= 0 s d=0 ft t= 0.5 s d=4 ft t= 1 s d=16 ft t= 1.5 s d=36 ft

5. A.How far does the clam in fall from t = 0 to t = .5 s? from t = .5 to t = 1 s? from t = 1 to t = 1.5 s? B.Is d a linear function of t? Why or why not? C.Graph you points from letter a and connect them with a smooth curve. Does your graph support your answer in b? Answer a, b, and c using the diagram on the left t= 0 s d=0 ft t= 0.5 s d=4 ft t= 1 s d=16 ft t= 1.5 s d=36 ft

6. Things to remember… • When you are writing a function from a word problem that deals with “free fall” we are ignoring air resistance and only dealing with gravity. • d(t) = 16t2 is the distance traveled by any falling object. In this function t is in seconds and d is in feet. • When solving quadratic functions if you aren’t able to solve the function by factoring you may have to use the quadratic formula

7. Suppose a gull drops a clam from a height of 50 ft. Express the height of the clam above the ground as a function of time. Graph the function and state the domain and range Use Example 1 Distance fallen 16t2 50 ft Height above ground h(t)

8. Example 2 • On the moon, the distance traveled (in meeters) by an object in freefall is given by s(t)= 0.8t2 , where t is the time (in seconds) that the object has been falling. • A. Suppose an astronaut tosses a small moon rock up and it reaches a height of 15 m before falling back to the surface of the moon. Express the height of the rock above the moon’s surface as a function of t, the time it is in freefall from a height of 15 m. • B. Graph the function in part A. What are the domain and range of the function?

9. The Height of a Thrown/Hit Object • If you throw an object instead of dropping it, the objects height in feet after t seconds is given by this equation: • h(t) = -16t2 + (vo sin A)t + ho • vo is the object’s initial speed in feet per second • ho is the object’s initial height in feet • If the units of measure are not in seconds and feet you have to convert them before you can use this equation!

10. Example 3 • When a baseball player hits a ball, the bat propels the ball from a height of 3 ft, at a speed of 150 ft/s, and at an angle of 53.10 with respect to the horizontal. • A. Express the height of the ball as a function of time. • B. When is the ball 59 ft above the ground?

11. Example 4 • A lacrosse player uses a stick with a pocket on one end to throw a ball toward the opposing team’s goal. Suppose the ball leaves a player’s stick from an initial height of 7 ft, at a speed of 90 ft/s, and at an angle of 300 with respect to the horizontal. • Express the height of the ball as a function of time • When is the ball 25 ft above the ground?

12. Summary of Today’s Lesson • Objects that are thrown/hit • Use h(t) = -16t2 + (vo sinA)t + ho • Remember units have to be seconds and feet • Objects that are dropped • Use h(t) = -16t2 + ho • Remember units have to be in seconds and feet • These two functions are the same except when you drop an object you don’t have an initial velocity, so in the second function the middle term is 0!

13. Homework • Pg 98/ 8, 9, and 10 parts a and b only