Problem: I have drawn a curve in the attachment. You are to print it out, and estimate the area in three different ways.

Procedure:
Part 1

a. Using your paper strips, cut them to fit outside the curved area. They should all be 1/2 inch in width, and you cut the length to fit. Line them up touching sides, with one end on the horizontal line at the bottom, and you cut the top end to fit just outside the curve. These paper strips must be rectangles. Don’t make any sloped cuts. Calculate the area of each rectangle and add them to find the estimated area of the parabolic figure. This area will be larger than the actual area.

b. Using your paper strips, cut them to fit inside the curved area. They should all be 1/2 inch in width, and you cut the length to fit. Line them up with one end on the horizontal line at the bottom, and you cut the top end, again, straight across. Calculate the area of each inscribed rectangle and add them to find the estimated area of the parabolic figure. This area will be smaller than the actual area.

c. Find the average of the two areas.

d. Thinking: The average area will be a closer approximation of the actual area of the parabolic figure. Let’s think about the new area. It’s the average of the two areas. Formulas: A=bh or base times height is the formula for the area of each original rectangle. That’s the same as length times width, right? Now if we take the base of the rectangles in the first estimate as b1 and the base of the rectangles in the second estimate as b2, the averageBase 1 plus Base 2 divided by 2 is then multiplied by h (height) we have A =Base 1 plus Base 2 divided by 2 multiplied by height. That formula should ring a bell. If not, get out a math book, and look for geometric figures, or area. This formula represents the area of what geometric figure?

e. Thinking: If the height (that’s the half inch measure) gets smaller and smaller, will the estimate get closer to the actual area?? That’s calculus. The height approaches zero and the number of figures used to estimate area approaches infinity. Interesting concept. Calculus is a wonderful tool!! There is no response needed for Part 1-e.

http://learn.flvs.net/educator/student/frame.cgi?tdove1*merraemmons*mpos=1&spos=0&option=hidemenu&slt=ob/dXrj28wKf.*2165*http://learn.flvs.net/webdav/educator_algebra2_v7/index.htm

Now lets use our Algebra II skills a bit.

Part 2: Let’s put an equation with that curve. Use y = x2-8x+7

a. What is the value of the discriminant? Review Lesson 3.05.
b. What types of roots will this equation have? Ex: two complex roots, one repeated root, two real rational roots or two real irrational roots? Review Lesson 3.05.
c. Find the roots by factoring and solving. Review Lesson 3.04.
d. Find the roots by using the quadratic equation. Review Lesson 3.04.
e. Find the roots by graphing and using the tracking tool. Click the icon below to use the MathResources Graphing Tool.

f. Find the y-intercept. Review Lesson 3.07.
g. Complete the square, and write that form of the equation. Review Lesson 3.07.
h. Find the vertex of the parabola. Review Lesson 3.07.
i. Can you think of a reason why this equation could not possibly be the equation of the curve you printed? Review Lesson 3.06. Answer all questions that are in bold print, and write a one paragraph conclusion.