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

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

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

Example 1 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. 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

Answer a, b, and c using the diagram on the left 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? 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

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) = 16t 2 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

Use Example 1 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 50 ft Distance fallen 16t 2 Height above ground h(t)

Example 2 On the moon, the distance traveled (in meeters) by an object in freefall is given by s(t)= 0.8t 2, 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?

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) = -16t 2 + (v o sin A)t + h o v o is the object’s initial speed in feet per second h o 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!

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 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?

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 30 0 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?

Summary of Today’s Lesson Objects that are thrown/hit Use h(t) = -16t 2 + (v o sinA)t + h o Remember units have to be seconds and feet Objects that are dropped Use h(t) = -16t 2 + h o 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!

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