pencil red pen highlighter GP notebook calculator Solve
pencil, red pen, highlighter, GP notebook, calculator Solve for x. a) b) use completing the square +1 +1 +1 Check! +1 +1 total: +1
During the next unit, we are going to discuss: What are conic sections? Where do they come from? Let’s take a look at how conics were discovered, defined, and how they are used today.
Conic sections were discovered by the ancient Greeks around 300 B. C, but these ideas continue to be used today. They are defined as follows: Conic sections - the _______ intersections obtained when _______ plane right–circular cones are cut by a _________
Depending on where the plane intersects the circular conical surface, there are 4 resulting 2–dimensional shapes. Label the 4 types of conic sections by the diagrams below: Parabola _____ Circle ______ Ellipse _____ Hyperbola _____
Let’s view a short animation illustrating how these four types of conic sections are found:
Due to time constraints this semester, we will not be able to study applications of conic sections, but in higher level math, you will see that conic sections are used in the following cases: Parabolas: shape of car headlights mirrors in reflecting telescopes television and radio antennae Ellipses: the orbit of planets around stars architecture of the U. S. Capitol and cathedrals
Hyperbolas: shadow of a lamp shade on a wall shock wave generated by the wing of supersonic plane 2 different hyperbolas By the way, if you're carrying a sharpened pencil (hexagonal prism type), you've got SIX hyperbolas on you.
Today, we are going to start with: (h, k) r points in a plane at a ____ circle - the collection of all ______ constant ______ fixed point, called the _______. center distance, from a _______ radius - a line segment with _____ endpoints that are the _____ center ______ point on the circle. of the circle and a ______ (x – h)2 + (y – k)2 = r 2 The standard form of a circle is __________, (h, k) and radius ____. r with center ______
1. Given the equation for a circle, identify its center and radius. b) a) (2, 3) r=5 (4, – 10) r = 12 c) (– 3, – 11) 2. Write the standard form a circle that satisfies each set of conditions. (x – h)2 + (y – k)2 = r 2 a) center (0, 4), r = 9 b) center (0, 0), r = 8 (x – 0)2 + (y – 4)2 = 92 x 2 + (y – 4)2 = 81 (x – 0)2 + (y – 0)2 = 82 x 2 + y 2 = 64 c) (10, 0), (x – 10)2 + (y – 0)2 = (x – 10)2 + y 2 = 5 2
3. Graph each circle. Identify its center and radius. a) b) (2, – 5) r=2 10 (0, 3) y r=4 10 y To graph a circle, graph the center first. Then graph at least four “easy” points that are “r” units away from the center. – 10 10 – 10 x
3. Graph each circle. Identify its center and radius. c) (0, 0) r=8 10 – 10 y 10 – 10
4. Write the standard form of a circle. Identify the center and radius. We need to complete the square “twice” for both the x’s and y’s. a) 9 1 center: (3, 1) r=4 b) 4 4 center: (– 2, 2) r=3
4. Write the standard form of a circle. Identify the center and radius. c) 16 25 center: (4, – 5) r=7 d) 9 36 center: (– 3, 6) r=8
5. Write the standard equation for each graphed circle. Be sure to read the scale carefully! a) 5 y b) 2 y x 5 – 5 center: (0, 0) r=4 50 y x 7 – 3 – 5 c) x 50 – 8 – 50 center: (3, – 4) r=3 center: (– 10, 10) r = 30
Your assignment is to finish the rest of the worksheets.
1. Given the equation for a circle, identify its center and radius. a) b) (0, 0) r=1 (0, 7) r=7 c) (15, 0) 2. Write the standard form a circle that satisfies each set of conditions. a) (– 7, – 18), r = 14 b) (– 2, 3), r = 2 (x + 2)2 + (y – 3)2 = 22 (x + 7)2 + (y + 18)2 = 142 (x + 7)2 + (y + 18)2 = 196 c) (12, 9), (x + 2)2 + (y – 3)2 = 4 (x – 12)2 + (y – 9)2 = 22
3. Graph each circle. Identify its center and radius. a) b) (1, – 1) r=1 10 – 10 (– 4, 0) y 10 10 – 10 r=5 x – 10 y 10 – 10 x
3. Graph each circle. Identify its center and radius. c) (– 2, – 2) r=7 10 – 10 y 10 – 10 x
4. Write the standard form of a circle. Identify the center and radius. a) 4 9 center: (2, – 3) r=5 b) +16 4 +16 16 4 16 center: (– 2, 4) r=6
5. Write the standard equation for each graphed circle. a) 2 y – 2 b) 16 x 12 y c) 9 y x – 6 4 – 16 – 12 center: (7, – 7) r=5 8 – 4 center: (– 6, 8) r=6 – 5 center: (2, 2) r=5 x
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II. Match the equation of the circle with its graph. A) D) B) E) C) F) 1) 2) 3) D F C 4) 5) 6) A B E
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