y = x^2 - 6*x + 8
Q01. Diagram
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Q02. y = x^2 - 6*x + 8
Properties of the curve
- y-intercept = 8
- zeros of y
- y = (x - 2)*(x - 4)
- Hence y = 0 when x = 2 and x = 4
- Asymptote
- None
- When x < 2
- Curve is decreasing
- range : y is positive
- When between x = 2 and x = 4
- There is vertex at (3, -1). It is minimum point
- y decreses from (0, 0) to (3, -1)
- y increase from (3, -1) to (4, 0)
- Range of y is negative
- When x > 4
- Curve is increasing
- Range of y is positive
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Q03. Find signs of y' and y" from curve
Signs of y'
- y' < 0 if x < 3
- y' = 0 if x = 3. The minimum point (3, -1)
- y' > 0 if x > 3
- y" > 0 for all x values
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Q03. Find y'
First derivative y'
- y = 2*x - 6
- y' < 0 if x < 3
- y' = 0 if x = 3
- y' > 0 if x > 3
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Q04. Find y"
Second derivative y"
- y' = 2*x - 6
- y" = 2. Hence curve is concave upward
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Q05. Fromula
y = a*x^2 + b*x + c
- Vertex : xv = -b/(2*a) and yv = F(xv)
- Discriminant : D = b^2 - 4*a*c
- Principal axis : x = xv
- Directrix
- Q = 1/(2*a)
- Q = distance from focus to a line perpendicular to principal axis
- Equation of directrix is y = yv - Q/2
- Focus
- xf = xv
- yf = yv + Q/2
- Three points method to sketch
- Point 1 : y-intercept
- Point 2 : 1st zero value
- Point 3 : 2nd zero value
- Two point methods to sketch
- Point 1 : y-intercept (0, c)
- Point 2 : The only one zero value (Vertex on x-axis)
- Point 3 : (x3, y3) symmetrical to (0, c) about principal axis
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