You identified, analyzed, and graphed quadratic functions. (Lesson 1 5) Analyze and graph equations of parabolas. Write equations of parabolas.

conic section degenerate conic locus parabola focus directrix axis of symmetry vertex latus rectum

Determine Characteristics and Graph For (y 3) 2 = 8(x + 1), identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.

For (x + 1) 2 = 4(y 2), identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola. A. B. vertex: ( 1, 2); focus: ( 1, 3); directrix: y = 1; axis of symmetry: x = 1 vertex: ( 1, 2); focus: ( 1, 1); directrix: y = 3; axis of symmetry: x = 1 C. D. vertex: ( 1, 2); focus: (0, 2); directrix: x = 2; axis of symmetry: y = 2 vertex: ( 1, 2); focus: ( 2, 2); directrix: x = 0; axis of symmetry: y = 2

Characteristics of Parabolas ASTRONOMY The parabolic mirror for the California Institute of Technology s Hale telescope at Mount Palomar has a shape modeled by y 2 = 2668x, where x and y are measured in inches. What is the focal length of the mirror?

ASTRONOMY The cross section of the image of a constellation can be modeled by 12(y 6) = x 2, where x and y are measured in centimeters. What is the focal length of the cross section? A. 6 centimeters B. 12 centimeters C. 3 centimeters D.

Write in Standard Form Write x 2 8x y = 18 in standard form. Identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.

Write in Standard Form

Write y 2 + 16x = 55 6y in standard form. Identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola. A. (y + 3) 2 = 16(x 4); vertex: (4, 3); focus: (0, 3); directrix: x = 8; axis of symmetry: y = 3 B. (y + 3) 2 = 16(x 4); vertex: ( 3, 4); focus: ( 3, 0); directrix: y = 8; axis of symmetry: x = 3 C. (y 3) 2 = 16(x 4); vertex: (4, 3); focus: (0, 3); directrix: x = 8; axis of symmetry: y = 3 D. (y 3) 2 = 16(x 4); vertex: (4, 3); focus: (8, 3); directrix: x = 0; axis of symmetry: y = 3

Write Equations Given Characteristics A. Write an equation for and graph a parabola with focus (2, 1) and vertex ( 5, 1).

Write Equations Given Characteristics B. Write an equation for and graph a parabola with vertex (3, 2) and directrix y = 1.

Write Equations Given Characteristics C. Write an equation for and graph a parabola that has focus ( 1, 7), opens up, and contains (3, 7).

Write an equation for and graph a parabola with focus ( 2, 5) and directrix x = 4. A. 12(x + 2) = (y 5) 2 B. 12(x 1) = (y 5) 2 C. 12(x 1) = (y 5) 2 D. 2(x + 2) = (y 4.5) 2

Find a Tangent Line at a Point Write an equation for the line tangent to y = x 2 2 at (2, 2).

Write an equation for the line tangent to y 2 = 4x + 4 at (0, 2). A. y = 2x + 2 B. y = x + 2 C. y = x + 2 D. y = 2x + 2

You analyzed and graphed parabolas. (Lesson 7 1) Analyze and graph equations of ellipses and circles. Use equations to identify ellipses and circles.

ellipse foci major axis center minor axis vertices co-vertices eccentricity

Graph Ellipses A. Graph the ellipse

Graph Ellipses

Graph Ellipses B. Graph the ellipse 4x 2 + 24x + y 2 10y 3 = 0.

Graph Ellipses

Graph the ellipse 144x 2 + 1152x + 25y 2 300y 396 = 0. A. C. B. D.

Write Equations Given Characteristics A. Write an equation for an ellipse with a major axis from (5, 2) to ( 1, 2) and a minor axis from (2, 0) to (2, 4).

Write Equations Given Characteristics B. Write an equation for an ellipse with vertices at (3, 4) and (3, 6) and foci at (3, 4) and (3, 2)

Write an equation for an ellipse with co-vertices ( 8, 6) and (4, 6) and major axis of length 18. A. B. C. D.

Determine the Eccentricity of an Ellipse Determine the eccentricity of the ellipse given by

Determine the eccentricity of the ellipse given by 36x 2 + 144x + 49y 2 98y = 1571. A. 0.27 B. 0.36 C. 0.52 D. 0.60

Use Eccentricity ASTRONOMY The eccentricity of the orbit of Uranus is 0.47. Its orbit around the Sun has a major axis length of 38.36 AU (astronomical units). What is the length of the minor axis of the orbit?

PARKS A lake in a park is elliptically-shaped. If the length of the lake is 2500 meters and the width is 1500 meters, find the eccentricity of the lake. A. 0.2 B. 0.4 C. 0.6 D. 0.8

Determine Types of Conics A. Write 9x 2 + 4y 2 + 8y 32 = 0 in standard form. Identify the related conic.

Determine Types of Conics B. Write x 2 + 4x 4y + 16 = 0 in standard form. Identify the related conic selection.

Determine Types of Conics C. Write x 2 + y 2 + 2x 6y 6 = 0 in standard form. Identify the related conic.

Write 16x 2 + y 2 + 4y 60 = 0 in standard form. Identify the related conic. A. B. 16x 2 + (y + 2) 2 = 64; circle C. D. 16x 2 + (y + 2) 2 = 64; ellipse

You analyzed and graphed ellipses and circles. (Lesson 7-2) Analyze and graph equations of hyperbolas. Use equations to identify types of conic sections.

hyperbola transverse axis conjugate axis

Graph Hyperbolas in Standard Form A. Graph the hyperbola given by

Graph Hyperbolas in Standard Form

Graph Hyperbolas in Standard Form B. Graph the hyperbola given by

Graph Hyperbolas in Standard Form

Graph the hyperbola given by A. B. C. D.

Graph a Hyperbola Graph the hyperbola given by 4x2 y2 + 24x + 4y = 28.

Graph a Hyperbola

Graph the hyperbola given by 3x2 y2 30x 4y = 119. A. B. C. D.

Write Equations Given Characteristics A. Write an equation for the hyperbola with foci (1, 5) and (1, 1) and transverse axis length of 4 units.

Write Equations Given Characteristics B. Write an equation for the hyperbola with vertices ( 3, 10) and ( 3, 2) and conjugate axis length of 6 units.

Write an equation for the hyperbola with foci at (13, 3) and ( 5, 3) and conjugate axis length of 12 units. A. B. C. D.

Find the Eccentricity of a Hyperbola

A. 0.59 B. 0.93 C. 1.24 D. 1.69

Identify Conic Sections A. Use the discriminant to identify the conic section in the equation 2x2 + y2 2x + 5xy + 12 = 0.

Identify Conic Sections B. Use the discriminant to identify the conic section in the equation 4x2 + 4y2 4x + 8 = 0.

Identify Conic Sections C. Use the discriminant to identify the conic section in the equation 2x2 + 2y2 6y + 4xy 10 = 0.

Use the discriminant to identify the conic section given by 15 + 6y + y2 = 14x 3x2. A. ellipse B. circle C. hyperbola D. parabola

Apply Hyperbolas A. NAVIGATION LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F than it is from station E. Find the equation for the hyperbola on which the ship is located.

Apply Hyperbolas

Apply Hyperbolas B. NAVIGATION LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F than it is from station E. Find the exact coordinates of the ship if it is 125 miles from the shore.

Apply Hyperbolas

NAVIGATION Suppose LORAN stations S and T are located 240 miles apart along a straight shore with S due north of T. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 60 miles farther from station T than it is from station S. Find the equation for the hyperbola on which the ship is located. A. B. C. D.

You identified and graphed conic sections. (Lessons 7 1 through 7 3) Find rotation of axes to write equations of rotated conic sections. Graph rotated conic sections.

Write an Equation in the x y -Plane Use θ = 90 to write x 2 + 3xy y 2 = 3 in the x y -plane. Then identify the conic.

Use θ = 60 to write 4x 2 + 6xy + 9y 2 = 12 in the x y -plane. Then identify the conic. A. B. C. D.

Write an Equation in Standard Form Using a suitable angle of rotation for the conic with equation x 2 4xy 2y 2 6 = 0, write the equation in standard form.

Write an Equation in Standard Form

A. B. C. D.

Write an Equation in the xy-plane

ASTRONOMY A sensor on a satellite is modeled by after a 60 rotation. Find the equation for the sensor in the xy-plane. A. B. C. D.

Graph a Conic Using Rotations

A. B. C. D.

Graph a Conic in Standard Form Use a graphing calculator to graph the conic section given by 8x2 + 5xy 4y2 = 2.

Use a graphing calculator to graph the conic section given by 3x2 6xy + 8y2 + 4x 2y = 0. A. B. C. D.

You modeled motion using quadratic functions. (Lesson 1 5) Graph parametric equations. Solve problems related to the motion of projectiles.

parametric equation parameter orientation parametric curve

Sketch Curves with Parametric Equations

Sketch the curve given by x = 2t 6 and y = t2 3 over 3 t 3. A. B. C. D.

Write in Rectangular Form Write y = 2t and x = t2 + 2 in rectangular form.

Write y = 4t 2 and x = 2t 4 in rectangular form. A. y = (x + 4)2 B. C. y = 2x + 8 D. y = x2

Rectangular Form with Domain Restrictions

A. y = (x 5)2, x < 5 B. y = (x + 5)2, x > 5 C. y = (x + 5)2, x < 0 D. y = (x 5)2, x > 0

Rectangular Form with θ as Parameter Write y = 5 sin θ and x = 3 cos θ in rectangular form. Then graph the equation.

Write y = 9 sin θ and x = 5 cos θ in rectangular form. Then graph the equation. A. B. C. D.

Write Parametric Equations from Graphs A. Use the parameter t = x 1 to write the parametric equations that can represent y = x 2 + 2. Then graph the equation, indicating the speed and orientation.

Write Parametric Equations from Graphs B. Use the parameter t = 2x to write the parametric equations that can represent y = x 2 + 2. Then graph the equation, indicating the speed and orientation.

Write Parametric Equations from Graphs

A. B. C. D.

Projectile Motion FOOTBALL Shane Lechler of the Oakland Raiders has the record career punting average with an average of 46.47 yards. Suppose that he kicked the ball with an initial velocity of 26 yards per second at an angle of 72. How far will the ball travel horizontally if he punts it with an initial height of 1 yard?

Projectile Motion

SOFTBALL Kensey hits a softball with an initial velocity of 83 feet per second at an angle of 34. How far will the ball travel horizontally if Kensey s bat was 3.5 feet from the ground at the time of impact? A. 103 ft B. 205 ft C. 255 ft D. 657 ft