Warm Up Please have your homework out and ready for me when I get to you. Find the vertex, AOS, maximum/minimum,y-intercept, and direction of opening of y = -x 2 – 6x – 10

Example #2:

You Do: Write the equation of a quadratic function with a vertex of (-3, -5), which passes through the point (1, 11) Write the equation of a quadratic function with a vertex of (-3, -5), which passes through the point (1, 11)

You Do 16 (1, 18)

Standard Form Vertex Form What is the vertex form of y = 2x x + 7 ? What is the vertex form of y = 2x x + 7 ? Find the x-coordinate of the vertex, then plug in to find the y-coordinate. Find the x-coordinate of the vertex, then plug in to find the y-coordinate.

You Do Write the function in vertex form. Write the function in vertex form. y = x 2 – 4x + 6

Hands Up, Pair Up You will need your notes, a pencil, and a sheet of paper. You will need your notes, a pencil, and a sheet of paper. When I tell you to do so, walk around the classroom high-fiving your classmates. When I tell you to do so, walk around the classroom high-fiving your classmates. When I say, pair up, the person who you are high-fiving will become your partner. When I say, pair up, the person who you are high-fiving will become your partner. Find a place to sit with your new partner. Find a place to sit with your new partner.

Quick Check 1. In your own words, define vertex and axis of symmetry of a parabola. 2. Write the equation of a quadratic function with a vertex of (-2, 9) and a y-intercept of Find the vertex, AOS, maximum/minimum, and y-intercept of y = 3x 2 – 4x – 2 When you are finished, turn it into the box.

Quadratic Functions… and their applications!

For a typical basketball shot, the balls height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t t +6. For a typical basketball shot, the balls height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t t +6. a) a) What is the maximum height of the ball? b) When will the shot reach the height of the basket? (10 feet)b) c) c) When will the ball hit the floor, if it missed the basket entirely?

a) What is the maximum height of the ball? Put it in your calculator! Put it in your calculator! Answer: The maximum height of the ball is 31 feet! Use your zooms and change your window until you see the maximum. Use your zooms and change your window until you see the maximum. Find the maximum! Find the maximum!

b) When will the shot reach the height of the basket? (10 feet) Key words to highlight: Key words to highlight: Put 10 in for y2 and find the… Put 10 in for y2 and find the… INTERSECTION! Answer: 2.4 seconds! When (so we are looking for our x) Height of the basket (10 feet)

c) When will the ball hit the floor, if it missed the basket entirely? What do we put in for y2? What do we put in for y2? y2 = 0 Now find the intersection! Now find the intersection! Answer: The ball will hit the floor after 2.64 seconds!

Example #2 The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. How long does it take for the diver to reach her maximum height after diving off of the platform? How long does it take for the diver to reach her maximum height after diving off of the platform?

Example #2 The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. What is the maximum height of the diver? What is the maximum height of the diver?

Example #2 The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. Find the time it takes for the diver to hit the water. Find the time it takes for the diver to hit the water.

YOU DO – write your answers on a sticky note: The height, H metres, of a rocket t seconds after it is fired vertically upwards is given by The height, H metres, of a rocket t seconds after it is fired vertically upwards is given by How long does it take for the rocket to reach its maximum height? How long does it take for the rocket to reach its maximum height? What is the maximum height reached by the rocket? What is the maximum height reached by the rocket? How long does it take for the rocket to fall back to earth? How long does it take for the rocket to fall back to earth?

The Chesapeake Bay

Average Monthly Temperatures of the Chesapeake Bay MonthJanFebMarAprMayJunJulAugSepOctNovDec Temp Turn on your STAT PLOT and Diagnostics (2 nd 0 x -1 ) 2. Enter your data in L1 and L2 3. Look at the data you have entered. What is the temperature doing? Now lets actually look at the STAT PLOT (Zoom 9). 4. Which function that weve studied would best model the data? Do a quadratic regression! STAT CALC 5

What is the r 2 value? r 2 =.927 This tells us that 92.7% of the time, the model is a good predictor, and the closer this value is to 1, the closer the data is to the model.

Analysis Analysis According to the model, what month does the maximum temperature occur? According to the model, what month does the maximum temperature occur? According to the model, during what months would the temperature be 50°? According to the model, during what months would the temperature be 50°? June! March and October

Darryl is standing on top of the bleachers and throws a football across the field. The data that follows gives the height of the ball in feet versus the seconds since the ball was thrown. Time Ht a. a. Show a scatter plot of the data. What is the independent variable, and what is the dependent variable? b. b. What prediction equation (mathematical model) describes this data? c. c. When will the ball be at a height of 150 feet? d. d. When will the ball be at a height of 100 feet? e. e. At what times will the ball be at a height greater than 100 feet? f. f. When will the ball be at a height of 40 feet? g. g. When will the ball hit the ground?

a. Show a scatter plot of the data. What is the independent variable, and what is the dependent variable? Independent variable (x): Time! (always!) Dependent variable (y): Height

b. What prediction equation (mathematical model) describes this data? QUADRATIC!!

c. When will the ball be at a height of 150 feet? Height (y) Height (y) Put 150 in y2. Put 150 in y2. What happened?!? Explain.

d. When will the ball be at a height of 100 feet? Put 100 in y2 and find the intersection! Put 100 in y2 and find the intersection!.34 seconds and 3.65 seconds

e. At what times will the ball be at a height greater than 100 feet?

f. When will the ball be at a height of 40 feet? 4.53 seconds

g. When will the ball hit the ground? g. When will the ball hit the ground? Put 0 in y2 and find the intersection! 4.98 seconds

Now try it on your own!

Journal #1 How are you doing in this class? What are you doing well on? What can you do to improve? What do you like about this class? What would you change? How are you doing in this class? What are you doing well on? What can you do to improve? What do you like about this class? What would you change? Write a paragraph (at least 5 sentences) and turn it into the box when you are finished. HOMEWORK: Finish the quadratic modeling worksheet.