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Pre-AP Algebra II – Chapter 10 Test Review Standards/Goals:

E.3.a.: I can identify conic sections (parabola, circle, ellipse, hyperbola) from their equations in standard form.

E.3.b.: I can graph circles and parabolas and their translations from given equations or characteristics with and without technology.

E.3.c.: I can determine the characteristics of circles and parabolas from their equations and graphs.

E.3.d.: I can identify and write equations for circles and parabolas from given characteristics and graphs.

G.GPE.1.: I can identify and recognize the properties of the conic sections o I can derive the equation of a circle, a parabola, an ellipse, and a hyperbola. o I can determine the domain of a conic section o I can derive the equation of a parabola o I can derive the equation of a circle when given the center and radius using the

Pythagorean Theorem o I can derive the equation of a circle by completing the square to find both the center

and the radius. o I can derive the equation of an ellipse, given its foci. o I can recognize the fact that the sum or differences of the distances from the foci is

constant. o I can derive the equation of a hyperbola using the foci.

G.GPE.2: I can derive the equation of a parabola given a focus and a directrix.

CIRCLES: #1. What is an equation of the circle at the origin and radius 12?

#2. What is the equation for the translation 𝑥2 + 𝑦2 = 16 eighteen units right and THREE units DOWN? What is the center?

#3. What is the equation for the translation 𝑥2 + 𝑦2 = 16 twelve units right and FOUR units UP? What is the center?

#4. What equation represents a circle with a center at (3, -8) and a DIAMETER of 10?

#5. What is the equation of the circle with center (7, -10) and radius 14

9?

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#6. A circle is tangent to the points (8, 0) and (0, 8) on the x and y axes. What is the equation of the circle? #7. A circle has a center of (17, -19) and an area of 81𝜋. What is the equation of the circle? COMPLETING THE SQUARE: #8. Use completing the square to determine the standard form of the circle and find the center and radius. Additionally, decide if the point (6, -5) lies on the circle or not.

𝑥2 + 𝑦2 − 4𝑥 − 6𝑦 − 3 = 0

COMPLETING THE SQUARE: #9. A circle with equation 𝑥2 + 𝑦2 = 41 has a center at the origin. If you shift the circle to the left 5 units and 7 units down, what is the equation of the new circle? What is the radius? Where is the new center?

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COMPLETING THE SQUARE: #10. Write the following circle in standard form. Additionally, state its radius and center. Determine whether the point (-7, 10) lies on the circle.

x2 + y2 + 16x – 8y = –72

COMPLETING THE SQUARE: #11. Write the following circle in standard form. Additionally, state its radius and center. Determine whether the point (8, -9) lies on the circle.

x2 + y2 – 10x – 4y = –20 #12.

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Identify the center and radius of each. Then, sketch its graph. #13. (𝑥 + 2)2 + (𝑦 + 2)2 = 25 #14. (𝑥 + 2)2 + (𝑦 − 1)2 = 9

#15. (𝑥 + 2)2 + (𝑦 − 2)2 = 9 #16. (𝑥 + 2)2 + (𝑦 − 4)2 =6

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#17.

COMPLETING THE SQUARE: #18. Use completing the square to determine the standard form of a parabola and to find the vertex, foci and directrix. Additionally, be able to know whether a parabola has a maximum or minimum.

3𝑥2 − 𝑦 + 6𝑥 + 15 = 0

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PARABOLAS: DIRECTRIX & FOCI (Parabolas). #19. Be able to write the equation of a parabola that has its vertex at the origin and a directrix in the following locations. SCENARIO #1: directrix of x = 6 SCENARIO #2: directrix of y = -8 #20. What is an equation of the parabola with the vertex at the origin and focus (0, 3)? COMPLETING THE SQUARE: #21. Complete the square to find the standard form of the following parabola. Additionally, state the vertex, whether it has a minimum or a maximum, its directrix, and its focus.

𝑥2 + 5𝑥 − 7 − 𝑦 = 0 FINDING STANDARD FORM & DIRECTRIX OF A HORIZONTAL PARABOLA: #22. What is CENTER of the parabola shown: (𝑦 − 5)2 = 12(𝑥 + 7) #23. What is the CENTER of the parabola with the equation: (𝑦 + 5)2 = 5(𝑥 − 7)

#24. What is the equation of a parabola that has its vertex at the origin and a directrix at x = 4?

a. 𝑦2 = −16𝑥 b. 𝑦2 = −4𝑥 c. 𝑥2 = −16𝑦 d. 𝑥2 = −4𝑦

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Ellipse:

COMPLETING THE SQUARE: #25. Use completing the square to determine the standard form of an ellipse.

x2 + 4y2 + 6x – 8y + 9 = 0

#26. A horizontal ellipse has the equation(𝑥 − 5)

25

2+

(𝑦 − 7)

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2= 1. What is a vertex?

#27. Write the following ellipse in standard form. Additionally, determine its center, x-intercepts, and coordinates of its foci.

4𝑥2 + 16𝑦2 = 64

#28. State the RANGE of the conic section shown: a. (-3, 3) b. (-5, 5) c. [-5, 5] d. [-3, 3] e. All real numbers

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#29.

#30.

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Hyperbola: #31. Consider this equation:

900(𝑥 + 9)2 − 300(𝑦 − 7)2 = 9000

a. What conic section is represented above?

b. What is the center of the conic section above?

c. Is the conic section above written in standard form? If not, write it in standard form.

d. What is ‘𝑎2’?

e. What is ‘𝑏2’?

f. What is ‘c’?

#32. A hyperbola has vertices (±2, 0) and one focus at (3, 0). What is the equation of the hyperbola in standard form?

#33. What is the center of the hyperbola with the equation (𝒚 − 𝟏)

𝟏𝟔

𝟐−

(𝒙 −𝟖)

𝟐𝟓𝟔

𝟐= 𝟏?

#34. Find the center:(𝒙 − 𝟐𝟎)𝟐

𝟓−

(𝒚+𝟏𝟏)𝟐

𝟔= 𝟏

#35. Consider the equation below. What conic section does it represent? What is its center? Write it in its standard form. Find its foci.

𝑥2 − 𝑦2 = 81

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DOMAIN/RANGE/INTERCEPTS: Identify the INTERCEPTS of each conic section. Additionally, give the DOMAIN and RANGE of each graph.

#4. Which conic section is shown shown?

a. Circle b. Parabola c. Ellipse d. Hyperbola

#5. Which conic section is shown shown?

a. Circle b. Parabola c. Ellipse d. Hyperbola

Which conic section do the following equations represent?

#1. 5𝑥2 + 4𝑦2 = 180 #2. 𝑦2

51=

63 − 𝑥2

63 #3.

𝑦2

51=

63+ 𝑥2

63