1. Arch Investigation B. Davis MathScience Innovation Center

2. Arch of Light B. Davis MathScience Innovatio Center Do the lights at this school cast parabolas on the wall? At the top of the stairs indirect lighting casts an interesting pattern on our walls. Is this shape a parabola or most likely another math shape?

3. Arch of Light B. Davis MathScience Innovatio Center Let’s Investigate! n Step one: take some photos n Select a clear photo for the investigation

4. Arch of Light B. Davis MathScience Innovatio Center Step 2: n We will superimpose our photo on some graph paper n The graph paper will help us to locate points on the curve n We need at least 3 points

5. Arch of Light B. Davis MathScience Innovatio Center Step 3: n We might adjust the picture to put it in a convenient place. n We must be careful not to distort the picture’s dimensions. n Record the 3 points.

6. Arch of Light B. Davis MathScience Innovatio Center Step 4: Results n the 3 points: n (1,1) n (5,3) n (13,2)

7. Arch of Light B. Davis MathScience Innovatio Center Step 4: Write system n Write 3 equations, one for each point:(1,1) (5,3) (13,2) n using quadratic equation: y = ax^2 + bx + c n 1: 1 = a + b + c n 2: 3 = 25a + 5b + c n 3: 2 = 169a + 13 b + c

8. Arch of Light B. Davis MathScience Innovatio Center Step 5: Change to Matrix Equation 1 = a + b + c 3 = 25a + 5b + c 2 = 169a + 13 b + c 1 1 1 25 5 1 169 13 1 abcabc 132132

9. Arch of Light B. Davis MathScience Innovatio Center Step 6: Solve Matrix Equation abcabc 1 1 1 25 5 1 169 13 1 132132 Inverse matrix is : Solution is: 1/48 - 1/32 1/96 -3/8 7/16 -1/16 65/48 -13/32 5/96 -5/96 13/16 23/96

10. Arch of Light B. Davis MathScience Innovatio Center Step 7: Write Math Model Solution is: Therefore the equation for the light is: a = - 5/96 b = 13/16 c = 23/96 y = ax^2 + bx + c y = -5/96x^2 + 13/16x + 23/96

11. Arch of Light B. Davis MathScience Innovatio Center Step 8: Regression analysis Enter 3 points into STAT EDIT Use STAT CALC 5 to find line of best fit Therefore the regression equation for the light is: With a correlation coefficient of: Y= -.05208x^2 +.8125x +.23985 1

12. Arch of Light B. Davis MathScience Innovatio Center Final Results The 3 points: n (1,1) n (5,3) n (13,2) lie on the line Y= -.05208x^2 +.8125x +.23985 The light hitting the wall could very well be a parabolic shape.

13. Arch of Light B. Davis MathScience Innovatio Center Step 9: Comparison and Conclusion When the decimals found using STAT CALC 5 are evaluated using VARS 5 EQ a MATH 1 VARS 5 EQ b MATH 1 VARS 5 EQ c MATH 1, it is found that they exactly match the a,b,c found using the matrix method to solve the system. With r =1, it is concluded that both methods give an accurate method of finding the line of best fit. The light hitting the wall could very well be a parabola.

14. Arch of Light B. Davis MathScience Innovatio Center Further Note: n Although the conclusion gives strong evidence that this curve is a parabola, further investigation may reveal that it is a hyperbola. The difference is whether or not the wall is exactly parallel to the axis of the cone of light. If it is even slightly off, a hyperbola (rather than a parabola) is the result.