1. Graphing & Word Problems September 23rd

2. Warm Up Use the quadratic formula to solve:

3. Graphing an Equation by hand What did we do before we had calculators? How did they graph? EASY! They used ordered pairs & Table!

4. Graphing an Equation by hand Plug in values for x, to find y f(x) = 2x - 5 Xf(x)Y(x, y) -22(-2) – 5 =- 9(-2,-9) 0 1 2

5. Graphing an Equation by hand Plug in values for x, to find y f(x) = ¾ x + 2 Xf(x)Y( x, y ) -4 -2¾ (-2) + 2 =½(-2, ½ ) 0 2 4

6. Graphing an Equation by hand Plug in values for x, to find y Xf(x)Y( x, y ) -3 -2 0 1 6

7. Graphing a QUADRATIC Equation by hand Plug in values for x, to find y Xf(x)Y( x, y ) -6 (-6) 2 +2(-6) -24= 0(-6,0) -4 -2 0 2 4 6 8

8. Graphing a QUADRATIC Equation by hand Plug in values for x, to find y Xf(x)Y( x, y ) -5 -3 0 1 2 3 5

9. Verifying Solutions Look at the graph—name 3 different solutions (x,y) to this equation. How do you know?

10. Verifying Solutions Plug the points you picked into the equation!

11. PLOT IT!!! I will give you and your partner and table & graph paper & equation Roll the dice 4 times For each number you land on, evaluate the equation at that number, put in the table, and plot on the graph Roll the dice 4 more times, evaluate the equation using the NEGATIVE of the number you rolled, put in a table, and plot on the graph If you land on the same number, ROLL AGAIN! We want 8 total x values (4 negative, 4 positive )

12. We can also use things we already learned to graph easily. VERTEX, ZEROS, and DIRECTION Step 1: Identify Direction of Opening Step 2: Find the Vertex Step 3: Plot the vertex Step 4: Find the zeros Step 5: Plot the zeros Step 6: Extend the graph with arrows

13. Graphing using Vertex, Zeros, and Direction of Opening

14. Now you can FINALLY use a calculator! Sometimes problems do not work out nicely (whole numbers) and we need to use a calculator to find important information about the problem. We already learned how to find the vertex and zeros on our calculator, now lets use them to answer questions!

15. WORD PROBLEMS---THE VERTEX X represents WHERE the maximum or minimum is located. Example: X is usually a form of “time”---it took 5 seconds for the ball to reach its maximum height Y represents the ACTUAL VALUE of the maximum or minimum. Example: Y is usually a form of “distance” or “height”---the ball reached a maximum height of 100 feet

16. WORD PROBLEMS---THE Y-INTERCEPT The Y-intercept represents the INITIAL HEIGHT of the object. Example: If the y-intercept is (0,5) that means the ball started 5 feet in the air, meaning someone could be holding it If it is (0,0) that means it starts on the ground! (zero height)

17. WORD PROBLEMS---ZEROS The ZERO or X-intercept represents when the object is on the ground. Example: If the x-intercept is (16,0) that means after 16 seconds the object is back on the ground

18. Example 1 Donelle hit a baseball into the air with an initial upward velocity of 38 feet per second. The height h in feet of the ball above the ground can be modeled by h = –16t² + 88t + 5, where t is the time in seconds after Donelle hit the baseball. What is the maximum height the ball reaches? How long does it take to reach the maximum height?

19. Example 2 A rocket is launched into the air. The height in feet reached over time in seconds is given by the function f(x) = -4x 2 + 58x + 250. What was the initial height of the rocket? ***What is the height of the rocket 10 seconds after take-off? (use what we learned today about ordered pairs!) After how many seconds will the rocket hit the ground?

20. Example 3 Jason jumped off of a cliff into the ocean in Acapulco while vacationing with some friends. His height as a function of time could be modeled by the function h(t)= −16t 2 + 480t +2, where t is the time in seconds and h is the height in feet. a.How long did it take for Jason to reach his maximum height? b. What was the highest point that Jason reached? c. Jason hit the water after how many seconds?

21. HOMEWORK Complete the worksheet. Graph the 4 equations BY HAND (which means DO NOT use a calculator) and complete the word problems. If you do not have a graphing calculator at home, find the information for the word problems you need NOW using one of my calculator.