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Warm-Up

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- 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

Sec 5.1.3

- Visualizing End Behavior of Graphs
- Deciphering How Many Solutions There Could/Should Be.

- 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

- 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.

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?

Rate of Change

- Solve the following system.
Check the graph.

Check the table.

- Solve the following system.
Check the graph.

Check the table.

- Solve the following system.
Check the graph.

Check the table.

- Solve the following system.
Check the graph.

Check the table.

- 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.

Intersection of a circle and a parabola

- Consider the following system:
Have many solutions are possible?

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.

- 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.

First solve the equation of the circle in terms of y.

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).

Choose choice #5 ZSquare

The two parts will not connect

Now use the Intersect key to find all points of intersection.

Did you notice both shapes are symmetrical about the

y-axis?

- Rearrange both equations
- Use equal values method.
- Rearrange and solve the quadratic.
- Sub y values into original equations and solve for x.

Final points of intersection:

(-4,3) (4,3) (-3,-4) (3,-4)

y = x2 + 3

x2+ y2 = 9

Use algebra before graphing!

- Review and Preview
- Page 230
- # 37-39, 41-43;

Hint for #42: Equal bases…

A= Algebraically

G=Graphing