Parabolas Name: ________________ Using quadratic models to identify functional extrema 1. Briefly recapitulate the argument made in class that of all rectangles with a given area, the square has the lowest perimeter-to-area ratio. You may explicitly refer to the numerical example from class. 2. Calculate the area of a circle with perimeter 100 m. For circles, the perimeter is usually called the circumference. Hint: calculate the radius first, and use the radius to calculate the area. Area: _____________ Calculate the difference in the circle’s and the square’s areas. Difference: ___________ Calculate the % difference in the circle’s area to the square’s area: the difference is what % of the square’s area? % difference: ___________ Alternately: the circle’s area is what % of the square’s area? 3. Percent: ___________ A projectile’s position, in meters, h(t), above the surface of the earth at time, t, in seconds is given by h(t) = ( –4.9)t2 + (5.1)t + 1.2 At what time is the height of the projectile a maximum? Explain your reasoning. Symmetry Time: ___________ My peculiar method of graphing parabolas depends heavily on symmetry. Symmetry is a powerful idea that has dominated subatomic and cosmological physics in the last 50 years. 4. Suppose the x-intercepts of a parabola are –3 and 7. Suppose further that the point (12, 23) is on the parabola. Graph this information (with labels) on quadrille paper. Without writing or calculating the equation of the parabola, give another point on the parabola and explain your reasoning. Graph and label this. Sketch the parabola. 5. Make (or steal) an example of a quadratic function. Calculate the axis of symmetry and y-intercept of its graph. Graph this information (with labels) on quadrille paper. Calculate, graph, and label the symmetric point of the y-intercept. Explain the process. Curve-fitting Every quadratic function is of the form, f(x) = ax2 + bx + c, where a is any non-zero real number and b and c can be any real numbers. 6. Suppose (1, 2); (–1, 3); and (2, 1) are on a parabola. Graph and label the points. The unknowns in this problem are the three coefficients, a, b, and c. The functional equation gives us three relationships that we can put into a system of linear equations and solve. In this case, the point (1, 2) means that f(1) = a(12)+ b(1) + c = 2 or a + b + c = 2. Use the other points to generate two more equations. Solve the system as in class and write f(x) with its numerical coefficients. Calculate the intercepts and vertex of the parabola. Sketch the graph. 7. Pick three non-collinear points with no two vertically aligned. Graph and label them. Use the method above to find the equation of the parabola through them. Calculate the intercepts and vertex of the parabola. Sketch the graph.