Warm up A parabola and a circle are graphed on the same set of axes. In how many ways can the two graphs intersect? Demonstrate the answer to each of the following by drawing a careful sketch. a)Can there be no POI? b)Can there be only one POI? c)Can there be exactly two POI? d)Can there be exactly three POI? e)Can there be exactly four POI? f)Can there be exactly five POI? g)What is the maximum number of POI possible.?

Intersection of a circle and a parabola Can you think of any other way?

HOW MANY SOLUTIONS? Sec 5.1.3

What is a system of equations? Answer: A system of equation just means 'more than 1 equation.'. A system of linear equations is just more than 1 line, see the picture:

what is the solution of 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.

How many solutions can systems of linear equations have? There can be zero solutions, 1 solution or infinite solutions--each case is explained in detail in the following slide Note: Although systems of linear equations can have 3 or more equations, we are going to refer to the most common case--a stem with exactly 2 lines.

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.

☺ 5-33a; page 229

☺ 5-33b; page 229

*** 5-33c; page 229

☺ 5-33d; page 229

Other systems Now consider the system of equations that consists of a line and a parabola i.e. a linear and a quadratic function. Generate a table similar to the one we just created, see slide #7. Next repeat the process for systems that consist of a two parabolas. Repast the process for systems that consist of a hyperbola and a circle.

*****5-34; page The next few slides will display how to use your calculators to solve the system of equations graphically. Go to slide number 25 to see the steps for algebraic method.

Graphing circles on your calculator 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.

Graphing circles on your calculator

Choose choice #5 ZSquare Square root is both positive and negative Looks like an oval… due to zooming.

The two parts will not connect The gap might even be more noticeable as the radius increases.

enter the equation of parabola Now use the Intersect key to find all points of intersection. Hint… both shape are symmetrical about the y axis. Careful when you use the Table of values… as you have three equations.

To make the graph easier to see, turn off half of the circle. De highlight the equal sign. Once one POI is found, use symmetry to find another POI. Points (4,3) and ( -4,3) are two of the answers.

Turn on the second half of the circle. Once one POI is found, use symmetry to find another POI. Points (3,-4) and ( -3,-4) are the other two answers.

In summary There are four points of intersection for this example: (4,3) and ( -4,3) (-3,-4) and ( 3,-4) Other systems may have less number of POI… see your notes for the warm up. The next slide solves the same system algebraically.

Algebraic method 1.Rearrange both equations to solve for a common variable. 2.Use equal values method. 3.Rearrange and solve the quadratic. 4.Sub each y value into original equations and solve for x.

On your own: Review your notes. Rewrite and fortify them if needed. Update your vocab list, if needed. Review and Preview Page 230 # 37-43; 39 is a milestone problem