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(one) Hyperbola (with two branches)

Parabola

Math 70: Parabola and Circle Objectives:

1) Given two points, calculate the coordinates of the midpoint using the midpoint formula

2

,2

2121 yyxxM . [Average the x-coordinates, average the y-coordinates.]

2) Given two points, calculate the distance between them using the distance formula

212

212 yyxxD [Pythagorean theorem, solve for c.]

3) Graph parabolas a. Study the equation’s structure first, determine if it’s up/down (chap 8) or left/right (chap 10)!

b. Review up/down parabolas which are functions khxay 2)( or cbxaxy 2

Vertex kh, or a

bh and )(hyk

0a opens up, 0a opens down

1a standard shape, 1a narrower than standard, 1a wider than standard

Equation of the axis of symmetry hx

c. Graph parabolas that open left or right and are not functions hkyax 2)(

Swap x and y!!!

Vertex kh, with h outside the parentheses, and k inside the parentheses (reversed!)

Vertex formula a

bk (y-coordinate!) and )(hxh

0a opens right, 0a opens left

1a standard shape, 1a narrower than standard, 1a wider than standard

Equation of the axis of symmetry ky (horizontal! Use the y-coordinate of vertex.)

4) Graph circles from equations in standard form 222 rkyhx , where the center is kh, and the

radius is r.

5) Re-write the general form of an equation of a circle 022 edycxbyax in standard form using

completing the square, twice! 6) Given the center and radius, write the equation of a circle in standard form.

Examples and Practice

1) Calculate the midpoint between

3

19,4 and

3

13,3

2) Calculate the distance between

3

19,4 and

3

13,3

3) Find the vertex, direction, axis of symmetry of 432

1 2 xy

4) Sketch graph of 2yx

5) Sketch graph of 412

1 2 yx

6) Use 542 2 yyx .

a. Find the vertex.

b. Rewrite the equation in the form hkyax 2)(

7) Sketch graph of 2513 22 yx

8) Sketch graph of 91 22 yx

9) Find the center and radius of 0168422 yxyx

10) Find the center and radius of 02

122 22 yx

11) Write the equation of a circle (in standard form) having center 3,7 and radius 3

2.

12) Use 262 yyx

a. Find the x-intercept b. Find the y-intercept.

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Name ~~~~~~~~~~~~~

Date ~~~~~~~~~~~~~-

Tl-84+ GC 34 Using GC to Graph Parabolae that are Not Functions of x

Objectives: Recall the square root property Practice solving a quadratic equation for y Graph the two parts of a horizontal parabola Identify the axis of symmetry and its equation

When solving for a variable or expression that has been squared, recall that there are usually two solutions. Example: x 2 = 9 can be factored to (x + 3)( x- 3) = O, giving solutions x = 3,-3 .

The square root property tells us to use ± when we take the square root of both sides of x 2 = 9.

Solve each of the following equations for y by isolating the square and using the square root property.

1) x = y 2 4) x=S(y-2)2-4

2) x = (y-2)2

5) x = -s(y- 2 )2 - 4

3) x=S(y-2)2

When we graph using the GC, we have only the Y= menu, not an X= menu. So we must have y as a function of x, not y 2 = an expression in x, or x = an expression in y. Solve for y first, using algebra. If the square root property was used, write each of the two functions.

6) Graph x = y 2 on the GC.

Step 1: Solve the equation for y using the square root property;

± J; = j;2 gives y = ±Fx.

Step 2: Write as two functions.

y 1 = Fx and y 2 = -Fx

Step 3: Enter these two functions into the GC Y= menu and graph.

[ ) CENTER]

CTD [tNT£RJ (ZOOM ) ITJ Copyright 2011 by Martha Fidler Carey. Pennission to reproduce is given only to current Southwestern College instructors and students.

Rev 8-8-13

Tl-84+ GC 34 Using GC to Graph Parabolae that are Not Functions of x Page 2

The vertex of a horizontal parabola can be found from the standard form of the equation, just as it was for a quadratic function. The roles of x and y are reversed. x,= a(y- k) 2 + h is the standard form of the equation, and (h,k) is the vertex.

- ", ... The axis of symmetry is a horizontal line through the vertex, with equation y = k . Notice that the y-coeffi.clent of.the.v.e~e~. k, is•inside the pa,tenthe~~. next to .they.. vari.able. • If a>O, the parabofa open·s in the positive x-direCtion, to the right. . lfia<O, the para.bo.la.opens·in the pegative x-direction, to the left.

7) Find the vertex of )(. ~ (y .;_). )2 Does' ~his parabola_.open left or right? , .. r "'·

8) Find the equation of the axis of symmetry.

• • , • ·"t-,·""· 4, ''.-·' ~ ... ~., 1 " ~· ,;. t - . ·'· . . . . ' . . 9) Solve x = (y'-2)2 for its two functions. Make a table on the.Ge for the two functions.

Step 1: Solve the equation for y using the square root property; (Use your previous work.)

Step 2: Write as two functions .

. · Step 3: Enter these two functions into the GC Y= menu and graph . . .

Step 4: Set up the GC table to begin at the vertex.

If the parabola opens left, you'll use the 0 key to move through the table.

If the parabola opens right, use 0 to move through the.table.:~·~ <t-~,. ·v;v-·-.: •_; ..... ~ . · . ~,. ') . J··~. ~ (_, Q.~ J (.GRAPH f-:: · .. . .. L ..,..,,.

Use the GC table to fill in this chart: x-value Y1=2+Fx Y2 = 2-.Fx 0 . ' ': . 1 4 ' '

9 ..

10)

·,I!' " ,. .• ./"' - ,.~··

. I -Notice that the x-values provided to you in the previous chart have a pattern. What is it?

Copyright 2011 by Martha Fidler Carey. Pennission to reproduce is given only to current Southwestern College instructors and students.

..

Rev 8-8-13

Tl-84+ GC 34 Using GC to Graph Parabolae that are Not Functions of x Page 3

11) Find the vertex of x = s(y- 2 )2 Does this parabola open left or right?

12) Find the equation of the axis of symmetry of x = s(y- 2 )2.

13) Solve x = s(y- 2 )2 for its two functions. Make a table on the GC for the two functions.

Step 1: Solve the equation for y using the square root property; (Use your previous work.)

Step 2: Write as two functions.

Step 3: Enter these two functions into the GC Y= menu and graph.

Steo 4: Set up the GC table to begin at the vertex. Use the GC table to fill in this chart:

x-value Y1 = Y2 = 0 5 20 45

14) Draw axes and label them with an appropriate scale. Plot the points and graph both of the curves, to get the horizontal parabola. Sketch the axis of symmetry.

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Copyright 2011 by Martha Fidler Carey. Pennission to reproduce is given only to current Southwestern College instructors and students.

Rev 8-8-13

Tl-84+ GC 34 Using GC to Graph Parabolae that are Not Functions of x, page 4

Graph. You may wish to use these steps: a. Does this parabola open left or right? b. What is the vertex? c. What is the equation of the axis of symmetry? d. Solve for y, write two functions. e. Make a table. f. Draw axes and label them with an appropriate scale. Plot the points and graph parabola.

Sketch the axis of symmetry.

15) Graph x = 5(y-2)2-4

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t[ji:J:{J}}:/-rt{}!=j:[:J:J_{}J:!J:J-t{j' Shortcut: The ordered pairs (x,y) that satisfy y =-5(x-2)2 -4 can be changed (swap x for y and y

for x) to get ordered pairs that satisfy x = -5(y- 2 )2 - 4. Because y = -5(x - 2 )2 -4 is already a function of x, no solving for y is needed. To use this shortcut, skip step d. above, and make a table

of values for y = -5(x - 2 )2 - 4. Then make a second table for graphing x = -5(y- 2 )2 - 4 by swapping x for y and y for x. If this shortcut doesn't make sense to you, don't use it. Copyright 2011 by Martha Fidler Carey. Pennission to reproduce is given only to current Southwestern College instructors and students.

Rev 8-8-13

Tl-84+ GC 34 Using GC to Graph Parabolae that are Not Functions of x, page 5

-~ Sraph. You may wish to use these steps: a. Does this parabola open left or right? b. What is the vertex? c. What is the equation of the axis of symmetry? d. Solve for y, write two functions. (Or use the shortcut) e. Make a table. f. Draw axes and label them with an appropriate scale. Plot the points and graph parabola.

Sketch the axis of symmetry.

16) Graph x = -5(y-2)2 -4

Copyright 2011 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

Rev 8-.8-13

Tl-84+ GC 34 Using GC to Graph Parabolae that are Not Functions of x, solutions, page 6

-" 1) y=±.£

_.----

2) y = 2±..Jx

3) y=2±Jf or y=2± ~ 4) Y = 2±~x+4 or Y = 2± ..J~5(-x+_4_)

5 5

5) y=2±~ or y~2±~-x5- 4 or y=2± ~-s~+ 4)

~~ V(0,2) a=1 >O, opens right. 8) y=2

TABLE SETUP Tb1St.art=0 .::i.Tbl=l

IndPnt: • Depend:

Ask Ask

9) y=2±..Jx, y1 =2+..Jx Y2 =2-.J; ~-------~cont x Y1 Y2

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X=0 x-value Y1=2+.J; Y2 =2-.J; 0 2 2 1 3 1 4 4 0 9 5 -1

10) The x-values are all perfect squares.

11) V(0,2)

12) y = 2

13) /x JS; /x /x y = 2 ± ~5 or y = 2 ±-

5-; y1 = 2 + ~5 , Y2 = 2 - ~5

Copyright 2011 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.