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The graph below represents a section of the curve y = x(x−2)

Mathematics

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Graph of y = x(x − 2)

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Key points of y = x(x − 2)
Pointxy
x-intercept00
Minimum1−1
x-intercept20

Step-by-step solution

8 steps

Here are the solutions to the problems you've provided:

1.i) Equation of the line CD

  1. Identify the coordinates of points C and D.

    From the given information, C is $(6, 0)$ and D is $(8, 0)$.

  2. Calculate the slope of the line CD.

    The slope mm is given by the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.

    m=0−08−6=02=0m = \frac{0 - 0}{8 - 6} = \frac{0}{2} = 0
  3. Write the equation of the line.

    Since the slope is 0, the line is horizontal. The equation of a horizontal line is y=ky = k, where kk is the y-coordinate of any point on the line. Both C and D have a y-coordinate of 0.
    The equation of the line CD is y=0\boxed{y = 0}.

    1.ii) Distance AB

  4. Identify the coordinates of points A and B.

    From the given information, A is $(0, 7)$ and B is $(0, 2)$.

  5. Calculate the distance between A and B.

    The distance dd between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by the formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

    dAB=(0−0)2+(2−7)2d_{AB} = \sqrt{(0 - 0)^2 + (2 - 7)^2} dAB=02+(−5)2d_{AB} = \sqrt{0^2 + (-5)^2} dAB=0+25d_{AB} = \sqrt{0 + 25} dAB=25d_{AB} = \sqrt{25}

    The distance AB is 5 units\boxed{5 \text{ units}}.

    2. Calculate the value of k

  6. Identify the given points and gradient.

    The points are (6,k)(6, k) and (k,10)(k, 10). The gradient mm is 2.

  7. Use the gradient formula to set up an equation for kk.

    The gradient formula is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.

    2=10−kk−62 = \frac{10 - k}{k - 6}
  8. Solve the equation for kk.

    Multiply both sides by (k−6)(k - 6):

    2(k−6)=10−k2(k - 6) = 10 - k 2k−12=10−k2k - 12 = 10 - k

    Add kk to both sides:

    3k−12=103k - 12 = 10

    Add 12 to both sides:

    3k=223k = 22

    Divide by 3:

    k=223k = \frac{22}{3}

    The value of kk is 223\boxed{\frac{22}{3}}.

    3.
    The image provides a sketch of the curve y=x(x−2)y = x(x-2) but does not state a specific question to be solved for this part. Therefore, I cannot provide a solution for question 3.

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