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Q01 |
- Diagram : Parabola and quadratic function
- Q02 | - Parabola and quadratic Function y = a*x^2 + b*x + c
- Q03 | - Quadratic function y = 0.5*x^2 - 3*x + 2
- Q04 | - Parabola y = 0.5*x^2 - 3*x + 2
- Q05 | - Copmpare y = x^2 and y = (x - h)^2 - k
- Q06 | - Reference
- Q02 | - Parabola and quadratic Function y = a*x^2 + b*x + c
Q01. Diagram :
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Q02. Parabola and quadrtic function y = a*x^2 + b*x + c
Quadratic function
- y-intercpt is c
- zeros of y is
- x = (-b + Sqr(b^2 - 4*a*c)/(2*a)
- x = (-b - Sqr(b^2 - 4*a*c)/(2*a)
- Vertex : x = -b/(2*a) and y = -(b^2 - 4*a*c)/(4*a)
- Range : form -(b^2 - 4*a*c)/(4*a) to + infinite
- Domain : all real x
- Curve
- It decrease from +infinite to yv
- It Increase from yv to +infinite
- It has minimum at (xv, yv)
- It concave upward
- Point p on curve. P to focus = P to directrix
- vertex at (-b/(2*a), -(b^2 - 4*a*c)/(4*a)
- Principal axis is x = -b/(2*a)
- Distance from focus to directrix is D = 1/(2*a)
- Focus at xf = (-b/(2*a), yf = D/2 - (b^2 - 4*a*c)/(4*a)
- Directrix is y = -b/(2*a) - D/2
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Q03. Quadrtic function y = 0.5*x^2 - 3*x + 2
Properties
- y-intercept is 1
- zeros of y is at x = 0.764 and x = 5.236
- Vertex (3, -2.5)
- y = 0.5*(x - 0.764)*(x - 5.236)
- The curve
- x < 0.764, y is positive and curve is decreasing
- x = 0.764, y is zero
- x > 0.764 and x < 3 y is negative and curve id decreasing
- x = 3.000 and y = -2.5 and the curve has minimum
- x > 3.000 and x < 5.236 and y is positive and curve is increasing
- x = 5.236, y is zseo
- x > 5.326, y is positive and curve is increasing
- x value ... Squence ... 1st diff ... 2nd diff
- ..... 1 ...... -0.5
- ..... 2 ...... -2.0 ....... -1.5
- ..... 3 ...... -2.5 ....... -0.5 ........ 1.0
- ..... 4 ...... -2.0 ....... +0.5 ........ 1.0
- ..... 5 ...... -0.5 ....... +1.5 ........ 1.0
- ..... 6 ...... +2.0 ....... +2.5 ........ 1.0
- etc
- y' = x - 3
- x < 3 and y' is negative and curve is decreasing
- x = 3 and y' is zero and curve has minimum
- x > 3 and y' is positive and curve is increasing
- y" = 1
- Curve is concave upward
- y" is same as the 2nd difference
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Q04. Parabola y = 0.5*x^2 - 3*x + 2
Question
- 1. Find Coordinates of vertex
- 2. Find distance from focus to directrix
- 3. Find Coordinates of focus
- 4. Find equation of principal axis
- 5. Find equation of directrix
- 1. Vertex
- xv = -b/(2*a) = -(-3)/(2*0.5) = 3
- yv = F(xv) = 0.5*(3)^2 - 3*(3) + 2 = -2.5
- 2. D = 1/(2*a) = 1/(2*0.1) = 1
- 3. Focus
- xf = xv = 3
- yf = yv + D/2 = -2.5 + 1/2 = -2
- 4. Equation of principal axis x = xv = 3
- 5. Equation of directrix y = yv - D/2 = -2.5 - 0.5 = -3
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Q05. Copmpare y = x^2 and y = (x - h)^2 - k
Comparison
- Similarity
- Both have D = 1/(2*a) = 1/2
- Both are parabola opening upward
- The curves are congruent
- Difference
- y = x^2
- Vertex at (0, 0)
- Focus at (0, 0.25)
- Equation of directrix y = -0.25
- y = (x - h)^2 - k
- Vertex at (h, k)
- Focus at (0, k + 0.25)
- Equation of directrix y = k - 0.25
- y = x^2
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Q06. Reference
Reference
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Subject |
Parabola
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