Intro to Conics Hyperbolas I Parts of a

Intro to Conics - Hyperbolas I. . Parts of a Hyperbola. A) Standard form: or B) The center is at (0 , 0). Or (h , k) if in (x–h)2 + (y–k)2 form. C) Vertices: goes in the direction of the + letter. 1) + x 2 goes sideways. 2) + y 2 goes up/down. 3) You also have “co-vertices” that go in the – letter direction. D) Parabolas: they face in opposite directions. 1) They go in the direction of the + letter. (+x 2=left/right, +y 2=up/down) E) Asymptote lines: hyperbolas have diagonal asymptote lines 1) They go through the corners of the “box” you draw using the vertices and co-vertices.

Intro to Conics - Hyperbolas II. Writing Equations of Hyperbolas in Standard Form. A) Get the x 2 & y 2 values on one side of the equal sign. 1) Move the # part to the other side of the equal sign. B) The number part must be equal to 1. 1) Divide (or multiply by reciprocal) to make it equal 1. C) There cannot be a number on the top with the x 2 or y 2. 1) You cannot change the 1 on the other side, so to get rid of the top number, we do a math trick. a) Flip the number attached to the letter and put it on the bottom of the fraction. b) The negative sign goes on the letter. NOT the #.

Intro to Conics - Hyperbolas Examples: Write these hyperbolas in standard form. 1) (divide by 10) (put a 10 on bottom) 2) (flip the # and put it on bottom) Put the minus sign on the top of the squared term or in front of it. or

Intro to Conics - Hyperbolas III. Finding the Foci & Asymptote Lines of Hyperbolas. A) Look at the numbers on the bottom of the x 2 & y 2 terms. B) The vertices go in the direction of the + term. 1) The “co-vertices go in the direction of the – term. C) Square root the bottom number to find the vertices and co-vertices. 1) These are distances from the center. D) Foci: are on the direction of the + term. 1) The foci are from the center. E) Asymptote lines: If you draw a rectangle using the vertices and “co-vertices” as the center of the edges of a box, the asy lines go thru the corners of the box. 1) The box & the asymptote lines are drawn with dotted lines.

Intro to Conics - Hyperbolas IV. Graphing Hyperbolas written in Standard Form. A) Find and graph the center of the hyperbola (0 , 0). 1) If in ± (x – h)2 ± (y – k)2 form, then the center is (h , k). B) Find and graph the vertices and “co-vertices”. 1) Square root the bottom #. 2) The + term is the vertices, the – term is the co-vertices. 3) From the center, graph dots for the vertices/co-vertices. C) Connect the dots with a dotted rectangular BOX. D) Draw the dotted asymptote lines thru the corner of the box. E) Draw opposite facing parabolas from the vertex and go towards, but don’t touch, the asymptote lines.

Intro to Conics - Hyperbolas Examples: Graph the hyperbolas and foci. State distances. 3) foci: x=± 3 y=± 4 4) foci: x=± 8 y=± 6