Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form.
a. xy + 1 = 0 Answer:
The x’y’-coordinate system has been rotated θ degrees from the xy-coordinate system. The coordinates of a point in the xy-coordinate system are given. Find the coordinates of the point in the rotated coordinate system.
a.Θ = 45o, (2, 1)
Answer: , None of the choices
The x’y’-coordinate system has been rotated θ degrees from the xy-coordinate system. The coordinates of a point in the xy-coordinate system are given. Find the coordinates of the point in the rotated coordinate system.
a.Θ = 90o, (0, 3)
Answer: (-3, 0), (0, -3), (3, 0), (-3, 0) A whispering gallery has a semielliptical ceiling that is 9 m high and 30 m long. How high is the ceiling above the two foci? Answer: 5.4 m Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. c. Answer:
Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form.
c. 9x2 + 24xy + 16y2 + 90x – 130y = 0
Answer:
The x’y’-coordinate system has been rotated θ degrees from the xy-coordinate system. The coordinates of a point in the xy-coordinate system are given. Find the coordinates of the point in the rotated coordinate system.
b. Θ = 30o, (1, 3) Answer: Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes.
b. xy – 2y – 4x = 0 Answer:
The orbit of a planet around a star is described by the equation where the star is at one focus, and all units are in millions of kilometers. The planet is closest and farthest from the star, when it is at the vertices. How far is the planet when it is closest to the sun? How far is the planet when it is farthest from the sun? Answer: 700 million km, 900 million km Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form.
5x2 – 6xy + 5y2 – 12 = 0
Answer:
A big room is constructed so that the ceiling is a dome that is semielliptical in shape. If a person stands at one focus and speaks, the sound that is made bounces off the ceiling and gets reflected to the other focus. Thus, if two people stand at the foci (ignoring their heights), they will be able to hear each other. If the room is 34 m long and 8 m high, how far from the center should each of two people stand if they would like to whisper back and forth and hear each other? Answer: 15 m Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes.
a. x2 – 2xy + y2 – 1 = 0 Answer:
Expand the expression in the difference quotient and simplify.
F(x) = x3 Answer: 3x2 + 3xh + h2, h ≠ 0 Expand the binomial by using Pascal’s Triangle to determine the coefficients.
(x + 2y)5 Answer: x5 + 10x4y + 40x3y2 + 80x2y3 + 80xy4 + 32y5 Use the Binomial Theorem to expand and simplify the expression.
(x2/3 - y1/3)3 Answer: X2 – 3x4/3y1/3 + 3x2/3y2/3 – y Use the Binomial Theorem to approximate the quantity accurate to three decimal places.
(1.02)8 Answer: 1.172 Use the Binomial Theorem to approximate the quantity accurate to three decimal places.
(2.99)12 Answer: 510,568.785 Expand the binomial by using Pascal’s Triangle to determine the coefficients.
(2t - s)5 Answer: 32t5 – 80t4s + 80t3s2 – 40t2s3 + 10ts4 – s5 Expand the expression in the difference quotient and simplify.
F(x) = √x Answer:
Find a quadratic model for the sequence with the indicated terms.
Answer:
Find Pk+1 for the given Pk.
Answer:
Calculate the binomial coefficient.
12C0
Answer: 1 Calculate the binomial coefficient.
(104) Answer: 210 Find a formula for the sum of the first n terms of the sequence.
Answer:
Find a quadratic model for the sequence with the indicated terms.
Answer:
Use the Binomial Theorem to expand and simplify the expression.
Use the Binomial Theorem to expand and simplify the expression.
(x + 1)4 Answer: x4 + 4x3 + 6x2 + 4x + 1 Use the Binomial Theorem to expand and simplify the expression.
2(x - 3)5 + 5(x - 3)2
Answer: 2x4 – 24x3 + 113x2 – 246x + 207 Use the Binomial Theorem to expand and simplify the expression.
(x2 + y2)4 Answer: x8 + 4x6y2 + 6x4y4 + 4x2y6 + y8 Find the specified nth term in the expansion of the binomial.
(10x – 3y)12, n = 9 Answer: 32,476,950,000x4y8 Find the specified nth term in the expansion of the binomial.
(4x + 3y)9, n = 8 Answer: 1,259,712x2y7 Find the sum using the formulas for the sums of powers of integers.
Answer: 70 Find the sum using the formulas for the sums of powers of integers.
Answer: 120 Find the specified nth term in the expansion of the binomial.
(x + y)10, n = 4 Answer: 120x7y3 Find the sum using the formulas for the sums of powers of integers.
Answer: -3402 Use the Binomial Theorem to expand and simplify the expression.
(y - 4)3 Answer: Y3 - 12y2 + 48y – 64 Write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a linear model, a quadratic model, or neither. The first item serves as your example. 1.
First six terms: 0, 3, 6, 9, 12, 15
First differences: 3, 3, 3, 3, 3 Second differences: 0, 0, 0, 0 Linear What are the coordinates of the center of the circle given by the equation x2+y2-16x-8y+31=0? Answer: (8,4)
An orbit of a satellite around a planet is an ellipse, with the planet at one focus of this ellipse. The distance of the satellite from this star varies from 300,000 km to 500,000 km, attained when the satellite is at each of the two vertices. Find the equation of this ellipse, if its center is at the origin, and the vertices are on the x-axis. Assume all units are in 100,000 km. 𝑥2 𝑦2 Answer: 16 + 15 = 1
What does r refer to in the following equation? (x-h)2+(y-k)2=r
Answer: The square of the radius of a circle.
A semielliptical tunnel has height 9 ft and a width of 30 ft. A truck that is about to pass through is 12 ft wide and 8.3 ft high. Will this truck be able to pass through the tunnel? Answer: No
Find the standard equation of the ellipse which satisfies the given conditions.
a. foci (-7,6) and (-1,6), the sum of the distances of any point from the foci is 14. (𝑥+4)2 (𝑦−6)2 Answer: 49 + 40 =1
Find the standard equation of the parabola with focus F(0, -3.5) and directrix y = 3.5. Answer: 𝑥 2 = −14𝑦
Find the standard equation of the ellipse which satisfies the given conditions. d. covertices (-4,8) and (10,8), a focus at (3,12) (𝑥−3)2 (𝑦−8)2 Answer: 49 + 65 =1 Determine the asymptotes of the equation: 𝑥2 𝑦2 − =1 16 20 √5 √5 Answer: 𝑦 = 𝑥 𝑎𝑛𝑑 𝑦 = − 𝑥 2 2
Find the standard equation of the parabola which satisfies the given condition: Vertex (1, -9), focus (-3, -9) Answer: (𝑦 + 9)2 = −16(𝑥 − 1)
Give the standard equation of the circle satisfying the given condition: center at the origin, radius 4Give the standard equation of the circle satisfying the given condition: center at the origin, radius 4. Answer: 𝑥 2 + 𝑦 2 = 16
What kind of symmetry does a circle have?
Answer: All of the answer choices are correct. Find the equation in standard form of the hyperbola whose foci are F1 (-4) and F2 (4), such that for any point on it, the absolute value of the difference of its distances from the foci is 8. 𝑥2 𝑦2 Answer: − =1 16 16
VEE MA. MIKAELA CZARINA DELA ROSA
Pre-Calculus Week 11-19
Use the value of the trigonometric function to evaluate the indicated functions. cos (-t) = -(1/5) i. sec(-t) Answer: -5 State the quadrant in which θ lies: Sin θ < 0 and cos θ < 0 Answer: Quadrant III Use the value of the trigonometric function to evaluate the indicated functions. sin t = 1/3 i. csc(-t) Answer: -3 Determine the quadrant in which each angle lies. The answer should be in the following format: ex. Quadrant I a. 130° Answer: Quadrant II
Determine two coterminal angles (one positive and one negative) for each angle.
Answer:
Find the period and amplitude.
Answer: Period: π, Amplitude: 3
Find the angle in radians.
Answer: 32/7 radians
Evaluate the sine, cosine and tangent of the real number. Write "undefined" if it's not possible. State the quadrant in which θ lies: Sin θ > 0 and tan θ < 0 Answer: Quadrant II
Find a and d for the function f(x) = a cos x + d such that the graph of f matches the figure.
Find the angle in radians.
Answer: 6/5 radians
Determine two coterminal angles (one positive and one negative) for each angle. Give your answer in degrees using the following format: ex. 34, -25
Answer: 324, -396
Find the period and amplitude. Answer: Period: π/5, Amplitude: 3
Find the period and amplitude.
Answer: Period: 1, Amplitude: ¼
Evaluate (if possible) the six trigonometric functions of the real number. If not possible, answer with "undefined."
