1. Investigate 1. Draw a dot on a piece of paper 2. Draw a second dot 3. Connect the dots with a straight line 4. Draw a third dot – draw as many straight lines that go through the dots as you can 1.You should have 3 lines 5. How many lines can you draw using 100 points?

2.  Try to find a relationship between the number of points and number of lines to predict how many lines we can draw with 100 points. Number of Points, x0123456 Maximum Number of Lines, y00 Number of Points, x0123456 Maximum Number of Lines, y001361015 Is this what you got?

3.  Let’s graph it: Number of Points, x0123456 Maximum Number of Lines, y001361015

4.  Is the curve approximately linear, quadratic or something else? ◦ Quadratic, since 2 nd differences are the same

5.  What are the zeros of the curve? ◦ Recall, zeros are the x-values when y = 0: ◦ Thus, zeros occur at (0,0) and (1,0) ◦ y = a(x-r)(x-s) becomes y = a(x-0)(x-1) = ax(x-1) ◦ We can use the point (2,1) to find ‘a’:  1 = a(2)(2-1)  1 = a(2)(1)  1 = 2a  a = 0.5, so y = 0.5x(x-1) Number of Points, x0123456 Maximum Number of Lines, y001361015

6.  Using the equation, y = 0.5x(x-1), how many lines can you draw with 100 points?  Translate to mathematical terms: what is y when x = 100? y = 0.5(100)(100-1) y = (50)(99) y = 4950 Therefore, you can draw a maximum of 4950 lines with 100 dots.

7. Data from the flight of a golf ball are shown. If the max height of the ball is 30.0m, determine an equation for a curve of good fit. Horizontal Distance (m)030608090 Height (m)0.022.030.027.022.5 We can draw the curve using the points

8. Shape of the curve seems to fit a parabola The vertex is the maximum height: 30.0m We can see one zero at (0,0) Where is the other zero? Since the parabola is a symmetric shape, the other zero must be 60 units to the right of the vertex, at x = 120

10. A competitive diver does a handstand dive from a 10m platform. The table of values below shows the time in seconds and the height of the diver, relative to the surface of the water, in meters. Time (s)00.30.60.91.21.5 Height (m)10.009.568.246.032.94-1.03 Determine an equation that models the height of the diver above the surface of the water during the dive.

11. We can assume that the maximum height (diving board height) is the vertex (10.00m). We can also estimate the value of the zeros. We see one zero must occur between 1.2 and 1.5 seconds, so estimate 1.4 seconds. Since a parabola is symmetric and vertex is at x=0, the other zero must be at -1.4. Time (s)00.30.60.91.21.5 Height (m)10.009.568.246.032.94-1.03

12. Time (s)00.30.60.91.21.5 Height (m)10.009.568.246.032.94-1.03

13.  Calculate or in some cases estimate the x- intercepts, zeros for a curve in the form y=a(x-r)(x-s)  The value of ‘a’ can be determined algebraically by substituting coordinates of a point (other than 0) that lies on or close to the line