Algebra 2 Unit 9 (Chapter 9)
0. Spiral Review
Worksheet 0
1. Find vertex, line of symmetry, focus and directrix of a parabola. (Section 9.2)
Worksheet 1 1 – 25
2. Find the center and radius of a circle. (Section 9.3)
Worksheet 2 1 – 26
3. Find the vertices and foci of an ellipse. (Section 9.4)
Page 637 3 – 15 odd, 35
Worksheet 3 1 - 13
4. Find the vertices, foci and asymptotes for a hyperbola. (Section 9.5)
Page 645 3 – 10, 15 – 17, 27 – 32
Worksheet 4 1 – 9
5. Classify conics, translate, and shade conics.
Worksheet 5A 1 – 17
Worksheet 5B 1 -10
Review
Worksheet 1 - 20
Worksheet 1
Example 1
Find the line of symmetry and vertex for y = 2(x – 1)2 + 3
Answer: 1) This is a parabola in descriptive form y = a( x – h)2 + k
2) Since ‘ a ‘ is a positive number it opens up.
3) The line of symmetry is found in the ( ), but you change the sign.
y = 2(x – 1)2 + 3
line of symmetry x = 1
The line of symmetry is the ‘fold line’ of the parabola.
The parabola is symmetric or balanced about the line of symmetry
4) The vertex is (h, k) y = 2(x – 1)2 + 3
y = a( x – h)2 + k
So the vertex is (1, 3)
Problems:
For each of the following,
a) find the line of symmetry
b) find the coordinates of the vertex
c) sketch the parabola
1. y = (x + 3)2 – 2 2. y = (x – 4)2 + 6 3. y = 4(x + 8)2
4. y = 2x2 + 5 5. y = – 3x2 6. y = – (x – 2)2
Example 2
Find the line of symmetry and vertex for y = x 2 + 6x + 8
Answer: 1) This is a parabola in standard form y = ax2 + bx + c
2) ‘ a ‘ is the number 1. Since it is a positive number it opens up
3) The line of symmetry is found by the formula x = –
The values are a = 1, b = 6, c = 8
So for this parabola we have x = – or x = – 3
4) The vertex is found by
( – , substituting into the original equation)
So for this problem the vertex is ( – 3, and plug –3 into x2 + 6x + 8)
(–3)2 + 6(–3) + 8
So the vertex coordinates are ( – 3, – 1)
Problems:
For each of the following,
a) find the line of symmetry
b) find the coordinates of the vertex
c) sketch the parabola
7. y = x2 + 8x + 13 8. y = x2 – 6x + 5 9. y = 2x2 + 16x + 3
10. y = –3x2 + 18x + 1 11. y = –x2 – 8x + 10 12. y = x2 + 4x – 3
13. y = 5x2 – 10x 14. y = x2 + 6x 15. y = – 2x2 – 8x
For each of the following,
a) find the line of symmetry
b) find the coordinates of the vertex
c) find the coordinates of the focus
d) find the directrix
e) sketch the parabola
16. y = x2 17. x2 = 4y 18. y = – x2
19. 2x2 = – 4y
20. For the parabola y = x2
a) find the line of symmetry
b) find the coordinates of the vertex
c) find the coordinates of the focus
d) find the directrix
e) sketch the parabola
21. Use the answers to #20 and translate to the parabola y = x2 + 4
a) find the line of symmetry
b) find the coordinates of the vertex
c) find the coordinates of the focus
d) find the directrix
e) sketch the parabola
22. Use the answers to #20 and translate to the parabola y = (x – 5)2
a) find the line of symmetry
b) find the coordinates of the vertex
c) find the coordinates of the focus
d) find the directrix
e) sketch the parabola
23. For the parabola y = x2
a) find the line of symmetry
b) find the coordinates of the vertex
c) find the coordinates of the focus
d) find the directrix
e) sketch the parabola
24. Use the answers to #23 and translate to the parabola y = x2 – 3
a) find the line of symmetry
b) find the coordinates of the vertex
c) find the coordinates of the focus
d) find the directrix
e) sketch the parabola
25. Use the answers to #23 and translate to the parabola y = (x + 3)2
a) find the line of symmetry
b) find the coordinates of the vertex
c) find the coordinates of the focus
d) find the directrix
e) sketch the parabola
26. Which of the following is a parabola?
a) y = 2x b) y2 = 2x2 c) y = 2x2 d) y = 2x3 e) y = 2
Worksheet 2
Give the center and radius of each circle. Sketch problems 1,2, and 3.
1. x2 + y2 = 49 2. (x – 2)2 + (y – 4)2 = 36 3. (x – 5)2 + y2 = 25
4. 2x2 + 2y2 = 8 5. x2 + (y + 5)2 = 6. (x + 4)2 + (y – 3)2 = 7
7. x2 + (y – 1)2 = 20 8. 4(x + 1)2 + 4(y + 2)2 = 16 9. x2 + y2 – 16 = 0
10. x2 + y2 – 6x = 0 11. x2 + y2 = 8y 12. x2 + y2 – 4x + 2y – 4 = 0
13. x2 + y2 + 10x – 4y + 20 = 0 14. x2 + y2 + 12x – 6y = 0
15. Write the equation of the circle that would result if the circle x2 + y2 = 4
is translated 3 units down and 2 units right.
Graph problems 16 and 17.
16. x2 + y2 49 17. x2 + (y – 2)2 25
18. Given: (x – 4)2 + y2 = 3
Is the origin’ inside’, ‘outside’ or ‘on’ the figure?
Identify the following as a circle, a parabola, or neither.
19. 5x2 + y = 7 20. 6x2 + 6y2 – 12 = 0 21. y2 = 16 – x2
22. y = 2x + 3 23. y = 2x2 + 3 24. 4y + 4x2 = 0
25. Find the vertex and line of symmetry for y = x2 + 10x + 15
26. Find the vertex, focus and directrix for y = x2 + 2
Worksheet 3
In problems 1 – 3, identify the vertices and foci of the
following ellipses. Sketch each.
1. 2.
3.
4. Sketch
5. Given:
Is the origin ‘inside’, ‘outside’, or ‘on’ the figure?
Put the following ellipses into standard form.
6. 4x2 + 9y2 – 24x – 90y + 225 = 0 7. 4x2 – 8x + y2 – 10y + 25 = 0
8. 16x2 + 25y2 + 32x – 150y = 159
Identify the following as a parabola, a circle or an ellipse.
9. x2 + y = 9 10. 4x2 + 16y2 = 16 11. 16x2 + 16y2 = 16
12. 6x2 + 12x + 9y2 – 18y – 20 = 0 13. 4x2 + 8y + 4y2 + 16y = 32
Worksheet 4
Find the asymptotes of the hyperbolas in problems 1 – 3.
1. 2. 3.
4. Graph: 5. Graph:
6. For problem #5, is the origin ‘inside’ or ‘outside’ the graph?
Write in standard form:
7. 4x2 + 8x – y2 = 0 8. 16x2 – y2 – 96x – 4y + 124 = 0
9. 4y2 – 9x2 – 54x + 40y – 17 = 0
Worksheet 5a
State whether the following is a parabola, a circle, an ellipse or a hyperbola.
1. x2 = 8y 2. 4x2 + 2y2 = 8 3. 3x2 + 3y2 = 81
4. 9x2 – 4y2 = 4 5. 3x2 + 4y2 + 8y = 8 6. 13x2 – 49 = – 13y2
7. y = x2 + 3x + 1 8. x2 – 4y2 + 10x – 16y = 5
9. (y – 4)2 + (x + 2)2 = 4 10. 3(x + 1)2 + (y – 2)2 = 9
11. Describe the translation of the graph of y = – 2x2 to the graph of
y = – 2(x + 10)2 (Did it move down 10, up 10, right 10 or left 10?)
12. Describe the translation of the graph of y = 3x2 to the graph of
y = 3x2 – 5 (Did it move down 5, up 5, right 5 or left 5?)
13. True or False: The equation is equivalent to the
equation
14. True or False: The equation is equivalent to the equation
15. True or False: The equation is equivalent to the
equation
16. Given: x2 + py2 – 4x + 10y – 26 = 0
Determine the shape of the following:
a. If p = 1
b. If p = 4
c. If p = – 4
17. Given: x2 + y2 – 10x + 10y + 1 = 0
Is the origin ‘inside’, ‘outside’, or ‘on’ the figure?
Worksheet 5B
Given the equation . If the graph is shifted down 2 units, which equation describes the new graph?
(a) (b) (c)
(d) (e)
2. Given . If the equation is shifted left 5 units, which equation describes the new graph?
(a) (b) (c)
(d) (e)
3. If the given function is shifted up 3 units and left 4 units, which equation describes the new graph?
(a) (b) (c)
(d) (e)
4. If the given function is shifted down 5 units, which equation describes the new function?
(a) (b) (c)
(d) (e)
5. If the graph of the equation is shifted to the right 3 units, which equation describes the new graph?
(a) (b) (c)
(d) (e)
6. Given . If the function is shifted 4 units to the left, write an equation that describes the new function?
7. Given . If the function is shifted 4 units down and 2 units right, write an equation that describes the new function?
8. Given . If the function is shifted 8 units to the right and 3 units up, write an equation that describes the new function?
9. The function may be formed by shifting the function in which way?
10. The function may be formed by shifting the function in which way?
Review
1. Describe the translation of the graph of y = 2x2 to the graph of
y = 2(x – 7)2
2. Describe the translation of the graph of y = 2x2 to the graph of
y = 2x2 – 7
3. How does the in y = (x + 1)2 + 3 change the graph of y = (x + 1)2 + 3
(move left, move right, move up, move down, make wider, make narrow, change
direction from opening up to opening down, change direction from opening down
to opening up)
4. Find the slope of the asymptotes of
In problems 5 – 9 determine if the equation represents a parabola, a circle, an ellipse or a hyperbola. Find the requested parts and sketch.
Parabola Circle Ellipse Hyperbola
line of symmetry center vertices vertices
vertex radius foci foci
focus asymptotes
directrix
5. x2 + (y – 4)2 = 9 6. y = (x + 5)2 + 3
7. 8. = 1
9. x2 + y2 – 10x + 6y – 15 = 0
10. Find the vertices 25(x + 3)2 + 4(y – 2)2 = 100
11. Find the vertices 4x2 – (y – 3)2 = 4
12. Find the line of symmetry and the vertex y = 2x2 + 12x + 23
For problems 13 – 15, identify the following conics and put into standard form.
13. x2 + y2 + 10x – 12y – 108 = 0
14. 9x2 – 16y2 + 54x + 32y – 79 = 0
15. x2 + 4y2 + 6x – 16y + 9 = 0
16. Find the vertex of y = 2x2 – 12x + 3
17. Graph: x2 + (y + 4)2 9
18. Is the origin ‘inside’ or ‘outside’ the graph for problem #17?
19. Graph:
20. Is the origin ‘inside’, ‘outside’ or ‘on’ the graph for problem #19?
Algebra 2 Unit 9 - 1 -