Objective: Students will understand what it means to describe, graph and write the equation of a parabola. Parabolas

Transcription

1 Pge 1 of 8 Ojective: Students will understnd wht it mens to descrie, grph nd write the eqution of prol. Prols Prol: collection of ll points P in plne tht re the sme distnce from fixed point, the focus F, nd fixed line, the directrix D d(f, P) = d(p, D) Eqution of Prol w/ (0, 0) & (, 0), > 0 is Vertex t (0, 0), > 0 Opens up Opens down Focus: Directrix: Opens right Focus: Directrix: Opens left Focus: Directrix: Focus: Directrix: ltus rectum - segment tht goes through the focus, nd its endpoints re points on the prol. The endpoints re distnce ±2 from the focus.

2 Pge 2 of 8 Ex 1 Discuss ech eqution. (So, find the vertex, focus nd directrix.) ) y 2 = 8x ) x 2 = -½y Ex 2 Find the eqution of the prol with vertex t (0, 0); xis of symmetry the x-xis; nd contins the point (2, 3). Vertex t (h, k) Use ptterns to descrie the focus nd directrix. Opens up Opens down Opens right Opens left

3 Ex 3 Find the eqution of the prol whose vertex is t (4, -2) nd focus is t (6, -2). Pge 3 of 8 Ex 4 Discuss the eqution: y y = -x + 1 You ve Got Prolems: Pge 623; 11-18, 19, 27, 29, 39, 47, 49,57, 55, 69

4 Ojective: Students will e le to grph, write the eqution, identify key elements nd convert etween forms. Ellipses Pge 4 of 8 Ellipse: the collection of ll points in the plne the sum of whose distnces from two fixed points, clled foci, is constnt Mjor xis the line contining the foci Center the midpoint of the segment joining the foci Minor xis the line tht s through the center nd perpendiculr to the mjor xis Vertices points t the intersection of the mjor xis nd the ellipse Co-vertices point t the intersection of the minor xis nd the ellipse x h yk Stndrd Form - : Generl form - x 2 + y 2 + cx + dy + e = 0. Ex 1 Lel the ellipse s mjor xis, center, minor xis, vertices nd co-vertices. Eqution of n Ellipse: (0, 0) (±c, 0) & (±, 0) (0, ±c) & (0, ±) where > > 0 nd 2 = 2 c 2 where > > 0 nd 2 = 2 c 2 FYI: Discuss the eqution in ellipses mens find the center, mjor xis, foci, vertices nd co-vertices. Ex 2 Grph the eqution of the conic section. Find the vertices, co-vertices nd foci y 9x 36

5 Pge 5 of 8 Ex 3 Find n eqution for ech ellipse given ) Center t (0, 0); focus t (-1, 0); vertex t (-3, 0) ) Foci t (0, ±2); mjor xis mesures 8 Eqution of n Ellipse: (h, k) (h ± c, k) & (h ±, k) (h, k ± c) & (h, k ± ) where > > 0 nd 2 = 2 c 2 where > > 0 nd 2 = 2 c 2 Ex 4 Find n eqution of n ellipse whose foci re (1, 2) nd (-3, 2) & whose vertex is (-4, 2). Ex 5 Discuss ech eqution. (So, find the center, foci, vertices, co-vertices.) ) 9(x 3) 2 + (y + 2) 2 = 18 ) 4x 2 + 3y 2 + 8x 6y = 5 How would you find the x-intercept of ny given ellipse? How would you find the y-intercept of ny ellipse? You ve Got Prolems: Pge 633: 13-16,25,27,29,39,49,55

6 Pge 6 of 8 Ojective: Students will e le to descrie, write the eqution of nd grph hyperols. Hyperols Hyperol: the collection of ll points in the plne the difference of whose distnces from two fixed points, clled foci, is constnt Trnsverse xis the line contining the foci Center the midpoint of the segment joining the foci Conjugte xis the line through the center & perpendiculr to the trnsverse xis Brnches two seprte curves which mke the hyperol s grph. They re symmetricl with respect to the trnsverse xis. Vertices points t the intersection of the mjor xis nd the hyperol Ex 1 Lel the xes, center, rnches, nd vertices of the hyperol. Eqution of Hyperol: (0, 0) (±c, 0) & (±, 0) Trnsverse xis prllel to x-xis (0, ±c) & (0, ±) Trnsverse xis prllel to y-xis where 2 = c 2 2 where 2 = c 2 2 Note: Asymptotes re y x. Note: Asymptotes re y x. FYI: Discuss the eqution for hyperols mens tht you ll find the center, trnsverse xis, foci, nd vertices tht hyperol. Rememer: How would you find the x-intercepts nd y-intercepts of hyperol?

7 Pge 7 of 8 Ex 2 Grph the hyperol. Then, find its center, foci, vertices nd symptotes. Eqution of Hyperol: (h, k) (h ± c, k) & (h ±, k) Trnsverse xis prllel to x-xis (h, k ± c) & (h, k ± ) Trnsverse xis prllel to y-xis where 2 = c 2 2 Note: Asymptotes re ( y k) ( x h). where 2 = c 2 2 Note: Asymptotes re ( y k) ( x h). Ex 4 Find n eqution for hyperol whose vertices re (4, 0) nd (-4, 0) nd hs n symptote of y = 2x. Then stte its foci Ex 5 Find n eqution for hyperol whose center is t (-3, 1), focus is t (-3, 6) nd whose vertex is t (-3, 4).

8 Pge 8 of 8 Ex 6 Find n eqution for hyperol whose vertices re t (1, -3) nd (1, 1) nd whose 3 symptote is y1 ( x 1). 2 You ve Got Prolems: Pge 647: 13-16, 17, 21, 27, 29, 31, 39