Algebra II Chapter 10 Conics Notes Packet. Student Name Teacher Name

Transcription

1 Algebra II Chapter 10 Conics Notes Packet Student Name Teacher Name 1

2 Conic Sections 2

3 Identifying Conics Ave both variables squared?' No PARABOLA y = a(x- h)z + k x = a(y- k)z + h YEs Put l'h squared!'erms together on the same side of fhe equal sign, Are both squ'aped 'ÿepms positive? HYPERBOLA (x - fo z (y - l,)z az b$ = 1 (y- toz (x-h)z l Are the coef,flclelats of the squared ferms equal? YES ELLIPSE (x-h)z (y-k)z az + ÿ--= I (x-h)z (y-k)z bz ÿ a-t---= 1 YEs CIRCLE r (x - h) Z + (y _ k) z = rz 3

4 E I 0J o II I X 0,b II H o +, >,, o tÿ o 0,l H, c4 W 4

5 Circles STANDARD FORM: CENTER: RADIUS: I. Rewrite in standard form. State the center and radius. Graph the circle. 1. x 2 + y 2 4x 16y + 64 = 0 2. x 2 + y 2 + 6x 2y 26 = 0 Center: Radius: Center: Radius: 5

6 Distance Formula: Midpoint Formula: II. Write the equation for the following circles in standard form. 1. Center ( 3, 2) and radius Center (2, 1) and goes thru point (5, 4). 3. Endpoints of the diameter are (10, 4) and (2, 4). III. Given the following circles, write the equation in standard form

7 Parabolas Notes 2 forms: 1. y = a(x-h) 2 + k Opens up or down. If a is positive, it opens up. If a is negative, it opens down. Vertex is at (h, k) 2. x = a(y-k) 2 + h Opens left or right. If a is positive, it opens to the right. If a is negative, it opens to the left. Vertex is at (h, k) 1 a 4 p p is the distance from the vertex to the focus. The AOS and the directrix are written as EQUATIONS!!!! The focus is located inside the parabola on the axis of symmetry. THE VERTEX IS HALFWAY BETWEEN THE FOCUS AND THE DIRECTRIX!!!!!!! 7

8 Directions for graphing a parabola: 1. Find and plot the vertex. 2. Decide how the parabola opens (up/down/left/right) 1 3. Find the p value. It is found by using the following: a. The p value is the 4 p distance from the vertex to the focus. (It is also the distance from the vertex to the directrix.) 4. Count and plot the focus. It is a point INSIDE the parabola 5. Count and plot the directrix. It is a line outside of the parabola. It NEVER touches the parabola. 6. Plot at least 2 points on each side of the vertex and sketch the parabola I. GRAPHING: 1. y 1 ( x 2) Opens Vertex a p Focus Directrix AOS 8

9 2. x 1 ( y 2) Opens Vertex a p Focus Directrix AOS 3. x 1 ( y 3) Opens Vertex a p Focus Directrix AOS II. Write the equation of each parabola with the given information. 1. Vertex (2, 3) and focus (0, 3) 2. Directrix: y = -5 and Focus (2, 1) 9

10 III. Write the equation of the parabolas below: Y Y X X IV. Rewrite each equation in vertex form. Fill in the blanks x y y y x x y 2 4x 4y 16 0 Vertex Vertex Vertex Opens Opens Opens a p a p a p Focus Focus Focus Directrix Directrix Directrix AOS AOS AOS 10

11 c 2 = a 2 b 2 (this helps find the foci) Ellipses always = 1 ELLIPSES NOTES HORIZONTAL VERTICAL Pictures: Standard Form: Center: Vertices: Co-Vertices: Foci: Major Axis: Minor Axis: Vertices: Co-Vertices: 11

12 Directions for graphing an ellipse: 1. Find and plot the center. 2. Decide if the major axis is vertical or horizontal. If the number under x 2 is the larger number, it is horizontal. If the number under y 2 is larger, it is vertical Take the square root of the larger number ( a ). Count that number of spaces from the center in the direction of the major axis Do the same for the minor axis except use the square root of the smaller number ( b ). 5. Sketch in the ellipse. 6. c 2 = a 2 b 2. This will help you find the foci(located inside the ellipse). The c value is the distance from the center to the foci. Count that number of spaces from the center and plot the foci. The foci are located on the major axis. 7. To find the coordinates of the foci, add and subtract the c value from the x- value of the center if the ellipse is horizontal and from the y-value if the ellipse is vertical. Write in standard form (if necessary). Find the center, a, b, c values, the vertices, co-vertices, foci, and the lengths of each axis. Then graph the ellipse. 1. ( x 4) ( y 3) Center: a=, b=, c= Vertices: Co-vertices: Foci: MA= ma= 12

13 2. ( x 2) ( y 2) Center: a=, b=, c= Vertices: Co-vertices: Foci: MA= ma= 2 2 ( x 1) ( y 3) Center: a=, b=, c= Vertices: Co-vertices: Foci: MA= ma= 13

14 4. 25x 2 + 9y 2 = 225 Center: a=, b=, c= Vertices: Co-vertices: Foci: MA= ma= 5. 25x y 2-50x 128y -119 = 0 Center: a=, b=, c= Vertices: Co-vertices: Foci: MA= ma= Write the equation for each ellipse. 14

15 6. Length of major axis is 14. Foci (4,0) and (-4, 0) 7. Vertices: (2, 8) and (2, 0). Co-vertices (5, 4) and (-1, 4). 8. MA endpoints (5, 10) & (5, 0); ma endpoints (3, 7) & (-1, 7) 15

16 Hyperbolas a 2 is ALWAYS under the POSITIVE term!!! a 2 + b 2 = c 2 This helps you find the foci 16

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19 Hyperbolas Day 2 Transverse Axis: Write the equation of the hyperbola that satisfies the given conditions. 2 Remember, you need the center, and values of a and b 2 1. Center (2, 2), transverse axis parallel to x-axis, a focus at (10, 2) and a vertex at (5, 2). 2. Center at (-2, 2), a vertex at (-2, -4), a focus at (-2, -6), transverse axis parallel to y-axis. 3. Foci at (4, 0) and (-4, 0), length of the transverse axis is 2 19

20 Write in standard form; then find all parts: x 4y 18x 24y x 4y 96x 40y

21 Classifying a Conic from its General Equation The graph of Ax 2 + Cy 2 + Dx + Ey + F = 0 is the following: 1. Circle: A = C 2. Parabola: AC = 0 A = 0 or C = 0, but not both 3. Ellipse: AC > 0 A and C have like signs 4. Hyperbola: AC < 0 A and C have unlike signs The test above is valid if the graph is a conic. The test does not apply to equations such as x 2 + y 2 = 1, whose graph is not a conic. 21

22 Identify the following equations as being linear, parabola, circle, ellipse, or hyperbola. 1. (x+3) (y 4)2 25 = 1 2. y = 2(x + 6) 3. 3x 4y = x 2 + 3y 2 = x 2 y x + 32 = ( x 2) y ( x 4) ( y 2)

23 Classify each conic section and write the equation in standard form. 1. y 2 + x + 8y 17 = 0 2. x 2 + y 2 + 6x 2y + 9 = x 2 + 4y 2 54x 8y 59 = y x 2 100x 125 = x 2 + 4y 2 20x 32y + 81 = 0 23