Unit 2 Conic sections Warm Up

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Presentation transcript:

Unit 2 Conic sections Warm Up List 3 things from the test yesterday that you think you may have struggled with and explain why?

History Conic sections is one of the oldest math subjects studied. The conics were discovered by a Greek mathematician named Menaechmus (c. 375-325 BC). Menaechmus’s intelligence was highly regarded… he tutored Alexander the Great.

History Appollonius Appollonius (c. 262-190 BC) wrote about conics in his series of books simply titled “Conic Sections”. Appollonious’ nickname was “the Great Geometer” He was the first to base the theory of all three conics on sections of one circular cone. He is also the one to assign the name “ellipse”, “parabola”, and “hyperbola” to three of the conic sections.

ELLIPSES

Ellipses Look like flat circles. The major axis is the longer one and the minor axis is the shorter one. Foci are always on the major axis. There are four vertices – two on the major axis and two on the minor axis.

Ellipses with centers at the origin: where a is always the larger number In the first equation, the major axis is on the x axis. In the second equation the major axis is on the y axis. How do we know this?

Ellipses Continued: C = the distance from the center (vertex) to the foci. It helps you find the coordinates of the foci

center: (0, 0) vertices: (0, 12) (0, -12) covertices: (6, 0) (-6, 0) h = 0 k = 0 a = 6 b = 12 vertices: (0, 12) (0, -12) covertices: (6, 0) (-6, 0) foci: (0, 10.4) (0, -10.4)

Ellipse Basically an ellipse is a squished circle (h , k) a b Standard Equation of an ellipse not at the origin: (h , k) a b Center: (h , k) a: horizontal axis-length from center to right/left edge of circle b: vertical axis- length from center to top/bottom of circle * You must square root the denominator

Ellipses Not at the origin Must always be 1 center: (h, k) If denominator of x is bigger If denominator of y is bigger a2 > b2 b2 > a2 major axis: horizontal major axis: vertical c2 = a2  b2 c2 = b2  a2 foci: (h ± c, k) foci: (h, k ± c)

GRAPH. Center: (2, 1) Vertices: (-1, 1) (5, 1) k=1 a=3 b=2 GRAPH. Center: (2, 1) Vertices: (-1, 1) (5, 1) Covertices: (2, -1) (2, 3) Focus: (-.2, 1) (4.2, 1)

GRAPH. Center: (2, 1) Vertices: (-1, 1) (5, 1) Focus: (-.2, 1) (4.2, 1) Covertex: (2, -1) (2, 3)

GRAPH. Center: (-3, -2) Vertices: (-3, -6) (-3, 2) k=-2 a=2 b=4 GRAPH. Center: (-3, -2) Vertices: (-3, -6) (-3, 2) Covertices: (-5, -2) (0, -2)) Focus: (-3, -5.5) (-3, 1.5)

GRAPH. Center: (-3, -2) Vertices: (-3, -6) (-3, 2) Focus: (-3, -5.5) (-3, 1.5) Covertex: (-5, -2) (0, -2))

Not in standard form, so complete the square GRAPH. 4 4 4 center: vertices: focus: covertex:

GRAPH. Center: (-3, 1) Vertices: (-5, 1) (-1, 1) k=1 a=2 b=1 GRAPH. Center: (-3, 1) Vertices: (-5, 1) (-1, 1) Covertices: (-3, 0) (-3, 2) Focus: (-4.7, 1) (-1.3, 1)

GRAPH. Center: (-3, 1) Vertices: (-5, 1) (-1, 1) Focus: (-4.7, 1) (-1.3, 1) Covertex: (-3, 0) (-3, 2)