Mon Warm-Up The following table shows the amount of money in a state pension fund from 1995 to Assuming the data follows a linear model, find the model that represents the data. Then predict how much money will be in the fund in the year Finally, use your model to predict when the fund will have a zero balance.

Mon Warm-Up 1995$48.2 Billion 1996$46.7 Billion 1997$45.2 Billion 1998$43.7 Billion 1999$42.2 Billion 2000$40.7 Billion 2001$39.2 Billion

Mon Warm-Up Solution f(t) = - 1.5t In 2025 the fund will have $3.2 Billion left The fund will run out of money sometime in 2028.

A Long Time Ago, On a Golf Course Far Away….

A golfer uses an 8 iron to hit his tee shot on a short par 3. A special strobe camera measures the height of the ball above the ground (assume the ground is level) for the first four seconds of its flight. Use the data below to find a linear function to model the height, y, of the golf ball after x seconds. 0 seconds0 feet 1 second94 feet

Use your function to find the height of the golf ball after 5 seconds, after 6 seconds and after 10 seconds.

After 5 seconds the ball is 470 feet high After 6 seconds the ball is 564 feet high After 10 seconds, the ball is 940 feet high Do these answers make sense?

Should we use this equation to model our data?

The following shows the ACTUAL height of the golf ball after x seconds. 0 seconds0 feet6 seconds84 feet 1 second94 feet6.5 seconds 39 feet 2 seconds156 feet6.6 seconds ft 3 seconds 186 feet6.7 sec ft 4 seconds184 feet6.8 sec.8.16 ft 5 seconds150 feet6.875 sec. 0 feet

This is what the graph of the function would look like if we plotted time as the independent, x, variable and height as the dependent, y, variable.

The figure graphed is called a parabola. The type of function that, when graphed, forms a parabola is called a Quadratic Function. All quadratic functions can be written in the form: Since a quadratic equation does not form a straight line, there is NO constant slope

Since our data more closely resembles a quadratic rather than a linear function, we would use the QUADRATIC REGRESSION model to find the function that best represents the data. Here is how we can come up with our Quadratic function

Let t represent the number of seconds that the ball is in the air, and h(t) will represent the height of the ball after t seconds. We will use the following 3 ordered pairs from our data.

So now we know that

This particular problem can be represented by the following quadratic function:

Homework Will be collected tomorrow Find the quadratic function that models each set of data

Warm Up A golfer is teeing off from a tee box that is 35 feet above the green.. One second after he tees off, the ball is 128 feet above the green and three seconds AFTER THAT, the ball is 23 feet above the green. Find the quadratic function that models the height of the golf ball above the green, as a function of time, t. How high above the green was the ball at the 2 second mark? At the 3 second mark? At the 5 second mark?

Warm Up Answer A golfer is teeing off from a tee box that is 35 feet above the green.. One second after he tees off, the ball is 128 feet above the green and three seconds AFTER THAT, the ball is 23 feet above the green. Find the quadratic function that models the height of the golf ball above the green, as a function of time, t. How high above the green was the ball at the 2 second mark? 157 ft. At the 3 second mark? 122 ft. At the 5 second mark? -140 ft.

Homework Answers Find the quadratic function that models each set of data

Use the graph to estimate: The maximum height of the golf ball, and The golf ball’s time of flight

We can calculate the ACTUAL maximum height. Remember from Algebra II, the x value at the vertex of a parabola is: Therefore, the maximum height reached by the golf ball is:

Therefore, the golf ball reached a maximum height of about 189 feet which occurred 3.44 seconds after it was struck.

Now, how do we find the total flight time of the ball?

Since parabola’s are symmetric, the time required to reach maximum height will be the same as the time required to return back to the ground.

Since it took seconds to go up, it stands to reason that it will take and additional seconds to come back down. Therefore, the total time of flight will be x 2 or seconds

Find a quadratic function that models the data Find the coordinates of the vertex Determine if the vertex is a maximum or minumum 1.

Find a quadratic function that models the data Find the coordinates of the vertex Determine if the vertex is a maximum or minumum 1.

Class Exercises Pg 442 #9-15 odd

Homework Pg 443 #49 – 52. Will be collected tomorrow

Wed. Warm Up The following equation models the height, h(t), of a bungie jumper t seconds after jumping off of a 100 foot high bridge. How close to the ground will he get? How many seconds after he jumps will he return to the bridge?

Wed. Warm Up Answer The following equation models the height, h(t), of a bungie jumper t seconds after jumping off of a 100 foot high bridge. How close to the ground will he get? 28 feet How many seconds after he jumps will he return to the bridge? 6 seconds

Homework Answers Pg 443 #49 – feet; 2 seconds feet; 12.5 seconds seconds; 81.6 meters feet; seconds

Wed. Classwork Do Quadratic Function worksheet. Finish for homework. Quiz tomorrow!

Thur. Classwork Quiz After quiz, do the worksheet “Choosing a model of best fit.” Finish for homework

Day 2 – Warm Up A U.S. Marine infantry squad in Iraq spots an insurgent ammunition cache 3 kilometers north of Baghdad. The Marines fire mortar rounds on the ammunition cache and the function below shows the height of each round, in feet, t seconds after the round is fired.

Day 2 – Warm Up a) How long does it take, to the nearest hundredth of a second, for each round to reach its maximum height? b) What is the maximum height of each round? c) How long is it from initial firing first impact?

Day 2 – Warm Up Answers a) How long does it take, to the nearest hundredth of a second, for each round to reach its maximum height? b) What is the maximum height of each round? c) How long is it from initial firing to first impact?

Matching Quadratic Functions to Data We can use a graphing calculator to find a quadratic function that models a data set. Input the data by going STATEDIT Find the regression by pressing STATCalc5QuadRegENTER

Find a quadratic function that models the data Find the coordinates of the vertex Determine if the vertex is a maximum or minumum 2.

Find a quadratic function that models the data Find the coordinates of the vertex Determine if the vertex is a maximum or minumum 3.

Find a quadratic function that models the data Find the coordinates of the vertex Determine if the vertex is a maximum or minumum 4.

Find a linear function that models the data Find the slope and the y intercept What model best fits the data? 5.

What model best fits the data below – Linear or Quadratic? Why? 6.

Since our data more closely resembles a quadratic rather than a linear function, we would use the QUADRATIC REGRESSION model to find the function that best represents the data. To get a quadratic equation that represents the data PressSTATCalc5QuadReg PressENTER