Q. What is the general form of the family of curves represented by y^2 = 4ax?
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A.
Parabolas opening to the right
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B.
Circles with varying centers
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C.
Ellipses with varying foci
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D.
Hyperbolas with varying asymptotes
Solution
The equation y^2 = 4ax represents a family of parabolas that open to the right with varying values of 'a'.
Correct Answer: A — Parabolas opening to the right
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Q. What type of curves does the equation (x^2/a^2) + (y^2/b^2) = 1 represent?
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A.
Ellipses
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B.
Circles
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C.
Parabolas
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D.
Hyperbolas
Solution
The equation (x^2/a^2) + (y^2/b^2) = 1 represents a family of ellipses with varying semi-major (a) and semi-minor (b) axes.
Correct Answer: A — Ellipses
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Q. What type of curves does the equation y = a + b cos(x) represent?
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A.
Linear functions
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B.
Cosine waves with varying amplitudes
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C.
Parabolas
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D.
Exponential functions
Solution
The equation y = a + b cos(x) represents cosine waves with varying amplitudes 'b' and vertical shifts 'a'.
Correct Answer: B — Cosine waves with varying amplitudes
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Q. What type of curves does the equation y = a e^(bx) represent?
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A.
Linear functions
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B.
Exponential functions
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C.
Trigonometric functions
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D.
Polynomial functions
Solution
The equation y = a e^(bx) represents a family of exponential functions with varying growth rates.
Correct Answer: B — Exponential functions
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Q. What type of curves does the equation y = a sin(bx + c) represent?
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A.
Linear functions
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B.
Exponential functions
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C.
Trigonometric functions
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D.
Polynomial functions
Solution
The equation y = a sin(bx + c) represents a family of trigonometric functions (sine waves) with varying amplitude (a) and frequency (b).
Correct Answer: C — Trigonometric functions
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Q. What type of curves does the equation y = a(x - h)^2 + k represent?
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A.
Linear functions
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B.
Parabolas
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C.
Circles
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D.
Ellipses
Solution
The equation y = a(x - h)^2 + k represents a family of parabolas with vertex at (h, k) and varying 'a' determining the direction and width.
Correct Answer: B — Parabolas
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Q. What type of curves does the equation y = e^(kx) represent?
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A.
Linear functions
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B.
Exponential functions
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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 with varying growth rates (k).
Correct Answer: B — Exponential functions
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Q. What type of curves does the equation y = k/x represent?
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A.
Hyperbolas
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B.
Parabolas
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C.
Circles
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D.
Ellipses
Solution
The equation y = k/x represents a family of hyperbolas where k is a constant.
Correct Answer: A — Hyperbolas
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Q. What type of curves does the equation y = kx^2 represent?
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A.
Straight lines
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B.
Parabolas with varying widths
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C.
Circles
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D.
Ellipses
Solution
The equation y = kx^2 represents a family of parabolas that open upwards or downwards depending on the sign of 'k'.
Correct Answer: B — Parabolas with varying widths
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Q. What type of curves does the equation y = mx^3 + bx + c represent?
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A.
Linear functions
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B.
Cubic functions
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C.
Quadratic functions
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D.
Exponential functions
Solution
The equation y = mx^3 + bx + c represents a family of cubic functions with varying coefficients.
Correct Answer: B — Cubic functions
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Q. What type of curves does the equation y = mx^3 + bx^2 + cx + d represent?
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A.
Linear functions
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B.
Quadratic functions
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C.
Cubic functions
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D.
Quartic functions
Solution
The equation y = mx^3 + bx^2 + cx + d represents a family of cubic functions with varying coefficients.
Correct Answer: C — Cubic functions
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Q. What type of curves does the equation y = mx^3 + c represent?
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A.
Linear functions
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B.
Cubic functions
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C.
Quadratic functions
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D.
Exponential functions
Solution
The equation y = mx^3 + c represents a family of cubic functions where m is the coefficient of x^3.
Correct Answer: B — Cubic functions
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Q. Which of the following is a family of exponential curves?
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A.
y = e^x
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B.
y = x^2
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C.
y = log(x)
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D.
y = sin(x)
Solution
The equation y = e^x represents a family of exponential curves for different bases.
Correct Answer: A — y = e^x
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Q. Which of the following is NOT a family of curves?
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A.
y = kx^2
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B.
y = ksin(x)
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C.
y = kx
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D.
y = k/x
Solution
y = kx represents a family of straight lines, but it is not a family of curves.
Correct Answer: C — y = kx
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Q. Which of the following represents a family of circles with varying radii?
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A.
(x - h)^2 + (y - k)^2 = r^2
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B.
(x - h)^2 + (y - k) = r
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C.
x^2 + y^2 = r
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D.
x^2 + y^2 = kx
Solution
The equation (x - h)^2 + (y - k)^2 = r^2 represents a circle centered at (h, k) with radius r.
Correct Answer: A — (x - h)^2 + (y - k)^2 = r^2
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Q. Which of the following represents a family of curves for the equation y = a sin(bx)?
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A.
Linear functions
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B.
Exponential functions
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C.
Sine waves with varying amplitudes and frequencies
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D.
Quadratic functions
Solution
The equation y = a sin(bx) represents sine waves where 'a' is the amplitude and 'b' is the frequency.
Correct Answer: C — Sine waves with varying amplitudes and frequencies
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Q. Which of the following represents a family of curves for the equation y = ax^2 + bx + c?
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A.
Linear functions
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B.
Quadratic functions
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C.
Cubic functions
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D.
Exponential functions
Solution
The equation y = ax^2 + bx + c represents a family of quadratic functions where 'a', 'b', and 'c' are constants.
Correct Answer: B — Quadratic functions
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Q. Which of the following represents a family of exponential curves?
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A.
y = ae^(bx)
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B.
y = ax^2 + bx + c
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C.
y = a sin(bx)
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D.
y = a log(bx)
Solution
The equation y = ae^(bx) represents a family of exponential curves where a and b are constants.
Correct Answer: A — y = ae^(bx)
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Q. Which of the following represents a family of straight lines?
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A.
y = mx + c
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B.
y = ax^2 + bx + c
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C.
y = e^x
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D.
y = sin(x)
Solution
The equation y = mx + c represents a family of straight lines for different values of m and c.
Correct Answer: A — y = mx + c
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