Q. What is the equation of an ellipse with foci at (±c, 0) and vertices at (±a, 0)?
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A.
x^2/a^2 + y^2/b^2 = 1
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B.
y^2/a^2 + x^2/b^2 = 1
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C.
x^2/b^2 + y^2/a^2 = 1
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D.
y^2/b^2 + x^2/a^2 = 1
Solution
The standard form of the equation of an ellipse with horizontal major axis is x^2/a^2 + y^2/b^2 = 1.
Correct Answer: A — x^2/a^2 + y^2/b^2 = 1
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Q. What is the equation of the circle with center (2, -3) and radius 4?
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A.
(x-2)² + (y+3)² = 16
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B.
(x+2)² + (y-3)² = 16
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C.
(x-2)² + (y-3)² = 16
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D.
(x+2)² + (y+3)² = 16
Solution
Equation of circle: (x-h)² + (y-k)² = r² => (x-2)² + (y+3)² = 4² = 16.
Correct Answer: A — (x-2)² + (y+3)² = 16
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Q. What is the equation of the circle with center (2, -3) and radius 5?
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A.
(x-2)² + (y+3)² = 25
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B.
(x+2)² + (y-3)² = 25
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C.
(x-2)² + (y-3)² = 25
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D.
(x+2)² + (y+3)² = 25
Solution
Equation of circle: (x-h)² + (y-k)² = r² => (x-2)² + (y+3)² = 25.
Correct Answer: A — (x-2)² + (y+3)² = 25
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Q. What is the equation of the circle with center (3, -2) and radius 5?
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A.
(x-3)² + (y+2)² = 25
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B.
(x+3)² + (y-2)² = 25
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C.
(x-3)² + (y-2)² = 25
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D.
(x+3)² + (y+2)² = 25
Solution
Equation of circle: (x-h)² + (y-k)² = r² => (x-3)² + (y+2)² = 5² = 25.
Correct Answer: A — (x-3)² + (y+2)² = 25
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Q. What is the equation of the directrix of the parabola x^2 = 8y?
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A.
y = -2
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B.
y = 2
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C.
x = -4
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D.
x = 4
Solution
The directrix of the parabola x^2 = 8y is y = -2.
Correct Answer: A — y = -2
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Q. What is the equation of the ellipse with center at the origin, semi-major axis 5, and semi-minor axis 3?
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A.
x^2/25 + y^2/9 = 1
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B.
x^2/9 + y^2/25 = 1
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C.
x^2/15 + y^2/5 = 1
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D.
x^2/5 + y^2/15 = 1
Solution
The equation of the ellipse is x^2/25 + y^2/9 = 1.
Correct Answer: A — x^2/25 + y^2/9 = 1
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Q. What is the equation of the line parallel to y = 2x + 1 that passes through the point (3, 4)?
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A.
y = 2x + 2
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B.
y = 2x + 1
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C.
y = 2x + 3
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D.
y = 2x - 2
Solution
Parallel lines have the same slope, so y - 4 = 2(x - 3) => y = 2x - 2.
Correct Answer: A — y = 2x + 2
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Q. What is the equation of the line parallel to y = 2x + 3 that passes through the point (1, 1)?
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A.
y = 2x - 1
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B.
y = 2x + 1
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C.
y = 2x + 3
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D.
y = 2x - 3
Solution
Parallel lines have the same slope: y - 1 = 2(x - 1) => y = 2x - 1.
Correct Answer: A — y = 2x - 1
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Q. What is the equation of the line parallel to y = 3x + 2 that passes through the point (1, 1)?
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A.
y = 3x - 2
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B.
y = 3x + 1
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C.
y = 3x + 2
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D.
y = 3x - 1
Solution
Parallel lines have the same slope, so y - 1 = 3(x - 1) => y = 3x - 1.
Correct Answer: D — y = 3x - 1
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Q. What is the equation of the line parallel to y = 3x + 4 that passes through the point (0, -2)?
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A.
y = 3x - 2
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B.
y = -3x - 2
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C.
y = 3x + 2
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D.
y = -3x + 4
Solution
Parallel lines have the same slope. The slope is 3, so using point-slope form: y + 2 = 3(x - 0) => y = 3x - 2.
Correct Answer: A — y = 3x - 2
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Q. What is the equation of the line parallel to y = 3x - 2 and passing through the point (2, 5)?
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A.
y = 3x + 1
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B.
y = 3x - 1
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C.
y = 3x + 2
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D.
y = 3x - 2
Solution
The slope of the given line is 3. Using point-slope form: y - 5 = 3(x - 2) gives y = 3x + 1.
Correct Answer: A — y = 3x + 1
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Q. What is the equation of the line parallel to y = 3x - 2 that passes through the point (2, 5)?
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A.
y = 3x + 1
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B.
y = 3x - 1
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C.
y = 3x + 2
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D.
y = 3x - 2
Solution
Since parallel lines have the same slope, the equation is y - 5 = 3(x - 2) which simplifies to y = 3x + 1.
Correct Answer: A — y = 3x + 1
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Q. What is the equation of the line parallel to y = 4x - 5 and passing through (2, 3)?
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A.
y = 4x - 5
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B.
y = 4x - 1
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C.
y = 4x + 5
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D.
y = 4x + 3
Solution
Parallel lines have the same slope. Using point-slope form: y - 3 = 4(x - 2) => y = 4x - 5.
Correct Answer: B — y = 4x - 1
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Q. What is the equation of the line parallel to y = 4x - 5 that passes through the point (2, 3)?
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A.
y = 4x - 5
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B.
y = 4x - 1
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C.
y = 4x + 5
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D.
y = 4x + 3
Solution
Parallel lines have the same slope. Using point-slope form: y - 3 = 4(x - 2) => y = 4x - 8 + 3 => y = 4x - 5.
Correct Answer: B — y = 4x - 1
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Q. What is the equation of the line parallel to y = 5x - 2 and passing through the point (2, 3)?
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A.
y = 5x - 7
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B.
y = 5x + 7
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C.
y = 5x - 2
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D.
y = 5x + 2
Solution
Parallel lines have the same slope. Using point-slope form: y - 3 = 5(x - 2) gives y = 5x - 7.
Correct Answer: A — y = 5x - 7
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Q. What is the equation of the line that is perpendicular to y = 3x + 1 and passes through the point (2, 3)?
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A.
y = -1/3x + 4
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B.
y = 3x - 3
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C.
y = -3x + 9
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D.
y = 1/3x + 2
Solution
The slope of the given line is 3, so the perpendicular slope is -1/3. Using point-slope form: y - 3 = -1/3(x - 2) gives y = -1/3x + 4.
Correct Answer: A — y = -1/3x + 4
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Q. What is the equation of the line that is perpendicular to y = 3x + 2 and passes through the point (2, 3)?
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A.
y = -1/3x + 4
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B.
y = 3x - 3
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C.
y = -3x + 9
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D.
y = 1/3x + 2
Solution
The slope of the perpendicular line is -1/3. Using point-slope form: y - 3 = -1/3(x - 2) gives y = -1/3x + 4.
Correct Answer: A — y = -1/3x + 4
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Q. What is the equation of the line that is perpendicular to y = 3x + 4 and passes through the origin?
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A.
y = -1/3x
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B.
y = 3x
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C.
y = -3x
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D.
y = 1/3x
Solution
The slope of the given line is 3. The slope of the perpendicular line is -1/3. Thus, the equation is y = -1/3x.
Correct Answer: A — y = -1/3x
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Q. What is the equation of the line that passes through the point (2, 3) and has a slope of -1?
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A.
y = -x + 5
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B.
y = -x + 3
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C.
y = x + 1
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D.
y = -x + 2
Solution
Using point-slope form: y - 3 = -1(x - 2) => y = -x + 5.
Correct Answer: A — y = -x + 5
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Q. What is the equation of the line with slope 2 passing through the point (1, 2)?
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A.
y = 2x + 1
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B.
y = 2x - 2
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C.
y = 2x + 2
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D.
y = 2x - 1
Solution
Using point-slope form: y - 2 = 2(x - 1) => y = 2x - 2 + 2 => y = 2x - 1.
Correct Answer: D — y = 2x - 1
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Q. What is the equation of the line with slope 3 passing through the point (1, 2)?
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A.
y = 3x + 2
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B.
y = 3x - 1
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C.
y = 3x + 1
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D.
y = 3x - 2
Solution
Using point-slope form: y - 2 = 3(x - 1) => y = 3x - 1.
Correct Answer: C — y = 3x + 1
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Q. What is the equation of the line with slope 3 that passes through the point (1, 2)?
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A.
y = 3x + 2
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B.
y = 3x - 1
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C.
y - 2 = 3(x - 1)
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D.
y = 2x + 1
Solution
Using point-slope form: y - y1 = m(x - x1) => y - 2 = 3(x - 1).
Correct Answer: C — y - 2 = 3(x - 1)
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Q. What is the equation of the line with slope 5 that passes through the point (1, 2)?
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A.
y = 5x - 3
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B.
y = 5x + 2
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C.
y = 5x + 1
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D.
y = 5x - 2
Solution
Using point-slope form: y - 2 = 5(x - 1) gives y = 5x - 3.
Correct Answer: C — y = 5x + 1
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Q. What is the equation of the parabola that opens upwards with vertex at the origin and passes through the point (2, 8)?
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A.
y = 2x^2
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B.
y = x^2
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C.
y = 4x^2
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D.
y = 8x^2
Solution
The vertex form of a parabola is y = ax^2. Since it passes through (2, 8), we have 8 = a(2^2) => 8 = 4a => a = 2. Thus, the equation is y = 4x^2.
Correct Answer: C — y = 4x^2
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Q. What is the equation of the parabola with focus at (0, 2) and directrix y = -2?
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A.
x^2 = 8y
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B.
x^2 = -8y
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C.
y^2 = 8x
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D.
y^2 = -8x
Solution
The distance from the focus to the directrix is 4, so the equation is y = (1/4)(x - 0)^2 + 0, which simplifies to x^2 = 8y.
Correct Answer: A — x^2 = 8y
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Q. What is the equation of the parabola with focus at (0, 3) and directrix y = -3?
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A.
x^2 = 12y
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B.
y^2 = 12x
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C.
y = 3x^2
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D.
x = 3y^2
Solution
The distance from the focus to the directrix is 6, so p = 3. The equation is y^2 = 4px = 12y.
Correct Answer: A — x^2 = 12y
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Q. What is the family of curves represented by the equation xy = c?
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A.
Hyperbolas
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B.
Parabolas
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C.
Ellipses
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D.
Circles
Solution
The equation xy = c represents a family of hyperbolas with varying constant c.
Correct Answer: A — Hyperbolas
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Q. What is the family of curves represented by the equation x^2/a^2 + y^2/b^2 = 1?
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A.
Ellipses
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B.
Hyperbolas
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C.
Parabolas
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D.
Circles
Solution
The equation x^2/a^2 + y^2/b^2 = 1 represents a family of ellipses with semi-major axis a and semi-minor axis b.
Correct Answer: A — Ellipses
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Q. What is the family of curves represented by the equation y = a sin(bx + c)?
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A.
Sine waves
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B.
Cosine waves
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C.
Linear functions
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D.
Quadratic functions
Solution
The equation y = a sin(bx + c) represents a family of sine waves with amplitude a and phase shift c.
Correct Answer: A — Sine waves
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Q. What is the family of curves represented by the equation y = e^(kx)?
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A.
Linear functions
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B.
Exponential functions with varying growth rates
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C.
Logarithmic functions
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D.
Polynomial functions
Solution
The equation y = e^(kx) represents a family of exponential functions where 'k' determines the growth rate.
Correct Answer: B — Exponential functions with varying growth rates
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