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A function is defined as f(x) = -x2 + 9. The points A and B are the x-intercepts, and C is the y-intercept. Find the length of AB, the coordinates of C, and the equation of the line passing through B

Mathematics

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3 parts
4.1

6 units6 \text{ units}

4.2
(0, 9)
4.3

-3x + 9

Graph of f(x) = −x² + 9

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Key points of f(x) = −x² + 9
Pointxy
x-intercept−30
Maximum09
x-intercept30

Step-by-step solution

9 steps

Okay 🥹❤️‍🔥M💞🥹Miseketeri, let's tackle Question 4.

Part 4.1· 2 steps

Determine the length of AB.

  1. Find the x-intercepts of f(x)f(x).

    The x-intercepts occur when f(x)=0f(x) = 0.

    −x2+9=0-x^2 + 9 = 0 x2=9x^2 = 9 x=±9x = \pm\sqrt{9} x=±3x = \pm 3

  2. Identify points A and B and calculate the length.

    From the diagram, A is the negative x-intercept and B is the positive x-intercept.
    So, A is (−3,0)(-3, 0) and B is (3,0)(3, 0).
    The length of AB is the distance between these two points.

    Length AB=3−(−3)=3+3=6\text{Length AB} = 3 - (-3) = 3 + 3 = 6

    The length of AB is 6 units\boxed{6 \text{ units}}.

Part 4.2(0, 9)

Determine the coordinates of C.

  1. Find the y-intercept of f(x)f(x).

    The y-intercept occurs when x=0x = 0.

    f(0)=−(0)2+9f(0) = -(0)^2 + 9 f(0)=9f(0) = 9

    The coordinates of C are (0,9)\boxed{(0, 9)}.

Part 4.3-3x + 9

Determine the equation of the straight line g(x)g(x).

  1. Identify two points on the line g(x)g(x).

    The problem states that g(x)g(x) passes through B and C.
    From 4.1, B is (3,0)(3, 0).
    From 4.2, C is (0,9)(0, 9).

  2. Calculate the slope (mm) of the line.

    m=y2−y1x2−x1=9−00−3=9−3=−3m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 0}{0 - 3} = \frac{9}{-3} = -3

  3. Determine the y-intercept (cc) and write the equation.

    The y-intercept is the y-coordinate of point C, which is 9.
    So, c=9c = 9.
    The equation of the straight line g(x)g(x) is g(x)=−3x+9g(x) = \boxed{-3x + 9}.

Part 4.4· 3 steps

Determine the distance of DF if OD = 5 units.

  1. Identify the coordinates of D.

    Given that OD = 5 units and D is on the x-axis, the coordinates of D are $(5, 0)$.

  2. Determine the coordinates of F.

    Assuming F is the point on the parabola f(x)f(x) with an x-coordinate of 5 (consistent with D(5,0) and DE being parallel to the y-axis).

    f(5)=−(5)2+9f(5) = -(5)^2 + 9 f(5)=−25+9f(5) = -25 + 9 f(5)=−16f(5) = -16

    So, the coordinates of F are (5,−16)(5, -16).

  3. Calculate the distance DF.

    Since D and F have the same x-coordinate, the distance DF is the absolute difference of their y-coordinates.

    Distance DF=∣0−(−16)∣=∣16∣=16\text{Distance DF} = |0 - (-16)| = |16| = 16

    The distance of DF is

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