Algebra 5/5/14 Bell Ringer 5 Minutes Time Remaining Algebra 5/5/14 Bell Ringer 5 Minutes 0:49 0:48 0:45 0:50 0:46 0:36 0:35 0:37 0:38 0:40 0:39 0:34 0:33 0:29 0:28 0:30 0:31 0:32 0:41 0:43 0:52 0:51 1:00 0:59 0:58 0:56 0:55 0:53 0:44 0:27 0:47 0:54 0:57 0:42 0:08 0:07 0:09 0:10 0:11 0:06 0:26 0:01 0:00 0:02 0:03 0:04 0:12 0:05 0:21 0:22 0:20 0:19 0:18 0:23 0:17 0:14 0:13 0:15 0:16 0:24 0:25 3:48 3:49 3:42 3:50 3:43 3:44 3:45 3:47 3:46 3:52 3:09 3:10 3:08 3:07 3:06 3:11 3:12 3:17 3:18 3:16 3:15 3:13 3:14 3:05 3:03 2:46 2:45 2:48 3:19 2:50 2:49 1:50 1:49 3:01 3:02 1:45 1:46 1:48 3:04 3:28 3:54 3:55 3:57 3:53 3:51 3:39 3:40 3:41 3:58 4:50 4:46 4:45 3:20 4:48 4:49 3:59 4:00 3:38 3:56 3:26 3:27 3:29 3:25 3:24 3:21 3:22 3:37 3:30 3:23 3:33 3:35 3:36 3:32 3:34 3:31 1:34 1:35 1:36 1:37 1:28 1:32 1:30 1:31 1:38 1:29 1:33 1:42 1:47 1:52 1:53 1:51 1:44 1:43 1:40 1:41 1:27 1:39 1:11 1:08 1:07 1:09 1:10 1:12 1:06 1:05 1:01 1:54 1:02 1:03 1:04 1:13 1:14 1:22 1:21 1:23 1:24 1:25 1:20 1:19 1:15 1:16 1:17 1:18 1:26 2:29 4:17 4:18 4:16 4:15 4:13 4:14 4:19 4:20 4:24 4:25 4:23 4:22 4:21 4:12 4:11 4:02 4:03 4:01 3:00 2:59 4:04 4:05 4:09 4:10 4:08 4:07 4:06 4:26 4:27 4:51 4:52 4:47 4:44 4:42 4:43 4:53 4:54 4:58 4:59 4:57 4:56 4:55 4:41 4:40 4:31 4:32 4:30 4:29 4:28 4:33 4:34 4:38 4:39 4:37 4:36 4:35 1:55 2:58 2:14 2:15 2:13 2:12 2:10 2:11 2:16 2:17 2:21 2:22 2:20 2:19 2:18 2:09 2:08 1:59 2:00 1:58 1:57 1:56 2:01 2:02 2:06 2:07 2:05 2:04 2:03 2:57 2:23 2:43 2:44 2:42 2:41 2:39 2:40 2:47 2:51 2:55 2:56 2:54 2:53 2:52 2:24 2:38 2:28 5:00 2:27 2:26 2:25 2:37 2:30 2:35 2:36 2:31 2:34 2:33 2:32 Evaluate 𝑏 2 −4𝑎𝑐 for the given quadratic equations. 𝑥 2 −2𝑥−9=0 𝑥 2 −8𝑥+16=0 −3𝑥 2 +4𝑥−5=0 Solutions 𝟒𝟎 𝟎 −𝟒𝟒

Applications of the Discriminant Objective: Use the discriminant to find the number of solutions of a quadratic equation. Vocabulary: Discriminant- within the quadratic equation, the discriminant is the expression inside of the radical.

So, Why do we care about the discriminant? The discriminant 𝑏 2 −4𝑎𝑐 of a quadratic equation can be used to find the number of solutions of a quadratic equation.

The Number of Solutions Of A Quadratic Equation Consider the equation 𝑎𝑥 2 +𝑏𝑥+𝑐=0. If 𝑏 2 −4𝑎𝑐 is positive, then the equation has two solutions. If 𝑏 2 −4𝑎𝑐 is zero, then the equation has one solution. If 𝑏 2 −4𝑎𝑐 is negative, then the equation has no solution.

Quick Check How many solutions does the given quadratic equation have? −𝑥 2 −5𝑥−9=0 4𝑥 2 −4𝑥+1=0 8𝑥 2 +8𝑥+1=0 −2𝑥 2 +3𝑥−5=0 Answers: No solutions One solution Two solutions

Challenging Questions– Think! If an equation has two solutions, how many x-intercepts does its graph have? If an equation has one solution, how many x-intercepts does its graph have? If an equation has no solution, how many x-intercepts does its graph have? Homework: p. 544 # 9-20

Parabolas Can Be Found In Some Unexpected Places…

Real- Life Representations of Quadratic Functions A ball was thrown and followed a path described by 𝑦=−0.02 𝑥 2 +𝑥. What was the maximum height (in feet) of the thrown ball? Let 𝑥 represent the distance traveled by the ball, and 𝑦 represent the height of the ball.

Real-Life Representation of Quadratic Functions On one of the banks of the Chicago River, there is a water cannon, called the Water Arc, that sprays recirculated water across the river. The path of the Water Arc is given by the model 𝑦=−0.006 𝑥 2 +1.2𝑥+10 where 𝑥 is the distance (in feet) across the river, 𝑦 is the height of the arc (in feet), and 10 is the number of feet the cannon is above the river. Questions: What is the maximum height of water sprayed from the Water Arc? How far across the river does the water land?

1. You throw a basketball whose path can be modeled by 𝒚=−𝟖 𝒙 𝟐 +𝟏𝟔𝒙+𝟔, where x represents time (in seconds) and y represents height of the basketball (in feet). What is the maximum height that the basketball reaches? How far across the room does the ball land?   2. A bottlenose dolphin jumps out of the water. The path the dolphin travels can be modeled by 𝒉=−𝟎.𝟐 𝒅 𝟐 +𝟐𝒅, where 𝒉 represents the height of the dolphin and 𝒅 represents horizontal distance. What is the maximum height the dolphin reaches? How far did the dolphin jump? 3. A model for a valley between two mountains whose peaks touch the x-axis is 𝒚= 𝟒𝟎.𝟒𝒙 𝟐 −𝟒𝟎𝟒𝒙, where 𝒙 and 𝒚 are measured in feet. How deep is the valley?

You will be asked to… Give your project a title. (2 pts.) Write down your scenario (2 pts.) Find the vertex. (5 pts.) What does the vertex represent? (5 pts.) Graph the model. (5 pts.) Is the vertex a maximum or a minimum point? (5 pts.) Use the quadratic formula to find the x-intercepts. (5 pts.) Does each x-intercept make sense in your problem? Why or why not? (5 pts.) Write your names on your project. (1 pt.) Bonus: Draw a picture to represent your scenario. (5pts.)

Algebra 5/6/14 Bell Ringer 5 Minutes Time Remaining Algebra 5/6/14 Bell Ringer 5 Minutes You are creating a rectangular banner for a school pep rally. You have 𝟏𝟎𝟎 𝒇𝒕 𝟐 (therefore, the area is 𝟏𝟎𝟎 𝒇𝒕 𝟐 ) of paper, and you want the length to be 𝟏𝟓 𝒇𝒕 longer than the width. What should be the dimensions of the banner? 0:46 0:45 0:50 0:48 0:49 0:39 0:40 0:37 0:38 0:34 0:30 0:29 0:28 0:31 0:32 0:35 0:42 0:33 0:36 0:47 0:55 0:56 0:54 0:53 0:52 0:57 0:58 0:27 0:44 0:51 1:00 0:59 0:43 0:41 0:08 0:01 0:09 0:11 0:12 0:02 0:07 0:26 0:05 0:06 0:04 0:03 0:13 0:10 0:21 0:20 0:00 0:23 0:25 0:24 0:19 0:22 0:15 0:14 0:16 0:17 0:18 3:49 3:48 3:47 3:45 3:42 3:43 3:44 3:46 3:52 3:50 3:08 3:09 3:07 3:06 3:05 3:10 3:11 3:16 3:17 3:15 3:14 3:12 3:13 3:04 3:03 2:49 2:48 2:50 1:49 1:50 2:46 2:45 3:18 3:01 1:45 1:46 1:48 3:02 3:28 3:54 3:55 3:57 3:53 3:51 3:39 3:40 3:41 3:58 4:50 4:46 4:45 3:19 4:48 4:49 3:59 4:00 3:38 3:56 3:25 3:26 3:27 3:24 3:23 3:20 3:21 3:37 3:29 3:22 3:33 3:35 3:36 3:32 3:34 3:31 3:30 1:34 1:33 1:35 1:36 1:28 1:31 1:30 1:37 1:29 1:32 1:42 1:44 1:47 1:51 1:52 1:43 1:27 1:39 1:40 1:41 1:38 1:10 1:08 1:07 1:09 1:11 1:12 1:06 1:05 1:01 1:53 1:02 1:03 1:04 1:13 1:14 1:22 1:21 1:23 1:24 1:25 1:20 1:19 1:15 1:16 1:17 1:18 1:26 2:29 4:17 4:18 4:16 4:15 4:13 4:14 4:19 4:20 4:24 4:25 4:23 4:22 4:21 4:12 4:11 4:02 4:03 4:01 3:00 2:58 2:59 4:04 4:05 4:09 4:10 4:08 4:07 4:06 4:26 4:27 4:51 4:52 4:47 4:44 4:42 4:43 4:53 4:54 4:58 4:59 4:57 4:56 4:55 4:41 4:40 4:31 4:32 4:30 4:29 4:28 4:33 4:34 4:38 4:39 4:37 4:36 4:35 1:54 2:57 2:13 2:14 2:12 2:11 2:09 2:10 2:15 2:16 2:20 2:21 2:19 2:18 2:17 2:08 2:07 1:58 1:59 1:57 1:56 1:55 2:00 2:01 2:05 2:06 2:04 2:03 2:02 2:22 2:23 2:42 2:43 2:41 2:40 2:38 2:39 2:44 2:47 2:54 2:55 2:53 2:52 2:51 2:56 2:37 2:27 2:28 2:26 2:25 2:24 5:00 2:30 2:34 2:35 2:36 2:33 2:32 2:31 Obj. SWBAT represent real life quadratic equation.

You will be asked to… Give your project a title. (2 pts.) Write down your scenario (2 pts.) Find the vertex. (5 pts.) What does the vertex represent? (5 pts.) Graph the model. (5 pts.) Is the vertex a maximum or a minimum point? (5 pts.) Use the quadratic formula to find the x-intercepts. (5 pts.) Does each x-intercept make sense in your problem? Why or why not? (5 pts.) Write your names on your project. (1 pt.) Bonus: Draw a picture to represent your scenario. (5pts.)

Algebra 5/7/14 Bell Ringer 5 Minutes Time Remaining Algebra 5/7/14 Bell Ringer 5 Minutes You are creating a rectangular banner for a school pep rally. You have 𝟓𝟎𝟎 𝒇𝒕 𝟐 (therefore, the area is 𝟓𝟎𝟎 𝒇𝒕 𝟐 ) of paper, and you want the length to be 𝟏𝟓 𝒇𝒕 longer than the width. What should be the dimensions of the banner? 0:46 0:45 0:50 0:48 0:49 0:39 0:40 0:37 0:38 0:34 0:30 0:29 0:28 0:31 0:32 0:35 0:42 0:33 0:36 0:47 0:55 0:56 0:54 0:53 0:52 0:57 0:58 0:27 0:44 0:51 1:00 0:59 0:43 0:41 0:08 0:01 0:09 0:11 0:12 0:02 0:07 0:26 0:05 0:06 0:04 0:03 0:13 0:10 0:21 0:20 0:00 0:23 0:25 0:24 0:19 0:22 0:15 0:14 0:16 0:17 0:18 3:49 3:48 3:47 3:45 3:42 3:43 3:44 3:46 3:52 3:50 3:08 3:09 3:07 3:06 3:05 3:10 3:11 3:16 3:17 3:15 3:14 3:12 3:13 3:04 3:03 2:49 2:48 2:50 1:49 1:50 2:46 2:45 3:18 3:01 1:45 1:46 1:48 3:02 3:28 3:54 3:55 3:57 3:53 3:51 3:39 3:40 3:41 3:58 4:50 4:46 4:45 3:19 4:48 4:49 3:59 4:00 3:38 3:56 3:25 3:26 3:27 3:24 3:23 3:20 3:21 3:37 3:29 3:22 3:33 3:35 3:36 3:32 3:34 3:31 3:30 1:34 1:33 1:35 1:36 1:28 1:31 1:30 1:37 1:29 1:32 1:42 1:44 1:47 1:51 1:52 1:43 1:27 1:39 1:40 1:41 1:38 1:10 1:08 1:07 1:09 1:11 1:12 1:06 1:05 1:01 1:53 1:02 1:03 1:04 1:13 1:14 1:22 1:21 1:23 1:24 1:25 1:20 1:19 1:15 1:16 1:17 1:18 1:26 2:29 4:17 4:18 4:16 4:15 4:13 4:14 4:19 4:20 4:24 4:25 4:23 4:22 4:21 4:12 4:11 4:02 4:03 4:01 3:00 2:58 2:59 4:04 4:05 4:09 4:10 4:08 4:07 4:06 4:26 4:27 4:51 4:52 4:47 4:44 4:42 4:43 4:53 4:54 4:58 4:59 4:57 4:56 4:55 4:41 4:40 4:31 4:32 4:30 4:29 4:28 4:33 4:34 4:38 4:39 4:37 4:36 4:35 1:54 2:57 2:13 2:14 2:12 2:11 2:09 2:10 2:15 2:16 2:20 2:21 2:19 2:18 2:17 2:08 2:07 1:58 1:59 1:57 1:56 1:55 2:00 2:01 2:05 2:06 2:04 2:03 2:02 2:22 2:23 2:42 2:43 2:41 2:40 2:38 2:39 2:44 2:47 2:54 2:55 2:53 2:52 2:51 2:56 2:37 2:27 2:28 2:26 2:25 2:24 5:00 2:30 2:34 2:35 2:36 2:33 2:32 2:31 Obj. SWBAT represent real life quadratic equation.

Quadratic Word Problems A picture has a height that is 4/3 its width. It is to be enlarged to have an area of 192 square inches. What will be the dimensions of the enlargement?

Quadratic Word Problems The product of two consecutive negative integers is 1122. What are the numbers?

Quadratic Word Problems A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway?

Quadratic Word Problems A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway?

Quadratic Word Problems A square and rectangle have the same area.  The length of the rectangle is five inches more than  twice the length of the side of the square.  The  width of the rectangle is 6 inches less than the  side of the square.  Find the length of the side of the square.

Quadratic Questions Which of the following represents the chart (input/output values) of a quadratic equation? 𝑥 𝑦 3 1 5 2 7 9 4 11 𝑥 𝑦 3 1 5 2 11 21 4 35