1. Warm-Up Visualize a circle and a parabola graphed on the same x-y plane Can you sketch the following scenarios? Zero points of intersection One point of intersection Two points of intersection Three points of intersection Four points of intersection Five points of intersection What is the maximum number of points of intersection we can have? Why?

2. Sec 5.1.3 HOW MANY SOLUTIONS?

3. Learning Targets Visualizing End Behavior of Graphs Deciphering How Many Solutions There Could/Should Be.

4. What is a solution for a system of equations? Answer: The solution is where the equations 'meet' or intersect. The red point on the graph is the solution of the system. This is where the two equations will produce the same output by using the same input

5. How many solutions can a system of linear equations have? There can be zero solutions, 1 solution or infinite solutions-- each case is explained on the following slide Note: Systems of equations can have 3 or more equations, but we are going to refer to a system with only 2 lines.

6. Case I: 1 Solution This is the most common situation and it involves lines that intersect exactly 1 time. Case 2: No Solutions This only happens when the lines are parallel. As you can see, parallel lines are not going to ever meet. Case 3: Infinite Solutions This is the rarest case and only occurs when you have the same line.

7. What Conclusions can we draw about intersecting lines? Come up with two conjectures as to what characteristics a system of equations has to have in order for there to be a solution. What does this say about the importance of the graphs end behavior?

8. P 229, 5-33a

9. P 229, 5-33b

10. P 229, 5-33c

11. P 229, 5-33d

12. Now consider the system of equations that consists of a line and a parabola i.e. a linear and a quadratic function. Next repeat the process for systems that consist of a two parabolas. Repeat the process for systems that consist of a parabola and a circle. Let’s Think About Other Systems

13. Intersection of a circle and a parabola

14. The next few slides will display how to use your calculators to solve the system of equations graphically. Go to slide number 18 to see the steps for algebraic method. Circle Example

15. A circle is not a function and cannot be graphed in the regular y=screen. To graph a circle in the regular y= screen, you have to graph it as two functions on the y= screen. Graphing Circles on your Calculator as Functions

16. Graphing Circles Cont.

17. Remember a square root can be positive or negative. In line 1 of y= screen graph what you've been graphing and then graph the same equation in line 2 but with a negative in front of the equation. You'll get something that looks like an oval since the calculator screen is rectangular. To make it look more circular (both parts aren't going to connect), press zoom and then select #5 (square). Graphing Circles Cont.

18. Choose choice #5 ZSquare The two parts will not connect

19. Now use the Intersect key to find all points of intersection. Did you notice both shapes are symmetrical about the y-axis? Graphing the System

20. 1.Rearrange both equations 2.Use equal values method. 3.Rearrange and solve the quadratic. 4.Sub y values into original equations and solve for x. Algebraic Method

21. Graph of System

22. You Solve y = x 2 + 3 x 2 + y 2 = 9 Use algebra before graphing!

23. Review and Preview Page 230 # 37-39, 41-43; On your own! Hint for #42: Equal bases…

24. Check in: You Should be Able to use the Following Methods for Solving Systems of Equations LinearQuadAbs ValCubicSq. RootCircleH-Bola LinearA/G GG G QuadA/G GGG G Abs ValGGGGGGG CubicGGGGGGG Sq. RootA/GGGGGGG CircleA/G GGGGG H-BolaGGGGGGG A= Algebraically G=Graphing