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Given y = -x + 4, determine the output value (y) when the input value is 9.

Mathematics

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Graph of y = −x + 4

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Key points of y = −x + 4
Pointxy
y-intercept04
x-intercept40

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QUESTION 8 [FUNCTIONS AND RELATIONSHIPS]

8.1.

  1. Substitute the input value x=9x=9 into the given equation y=−x+4y = -x + 4.

    y=−(9)+4y = -(9) + 4
  2. Perform the subtraction.

    y=−9+4y = -9 + 4 y=−5y = -5

    The output value is −5\boxed{-5}.

    8.2.

  3. To find aa, substitute x=4x=4 into the rule y=2x+1y = 2x + 1.

    a=2(4)+1a = 2(4) + 1 a=8+1a = 8 + 1 a=9a = 9
  4. To find bb, substitute x=5x=5 into the rule y=2x+1y = 2x + 1.

    b=2(5)+1b = 2(5) + 1 b=10+1b = 10 + 1 b=11b = 11

    The values are a=9a = \boxed{9} and b=11b = \boxed{11}.

    8.3.

  5. Examine the input and output values to find a pattern.

    Inputs: 2, 3, 8, 10
    Outputs: 16, 22, 52, 64

  6. Test a linear relationship of the form y=mx+cy = mx + c.

    Using the first two pairs (2, 16) and (3, 22):
    For (2, 16): 16=2m+c16 = 2m + c (Equation 1)
    For (3, 22): 22=3m+c22 = 3m + c (Equation 2)

  7. Subtract Equation 1 from Equation 2 to find mm.

    (22−16)=(3m+c)−(2m+c)(22 - 16) = (3m + c) - (2m + c) 6=m6 = m
  8. Substitute m=6m=6 into Equation 1 to find cc.

    16=2(6)+c16 = 2(6) + c 16=12+c16 = 12 + c c=16−12c = 16 - 12 c=4c = 4
  9. Formulate the rule and verify with other input-output pairs.

    The rule is y=6x+4y = 6x + 4.
    Check for x=8x=8: y=6(8)+4=48+4=52y = 6(8) + 4 = 48 + 4 = 52 (Matches)
    Check for x=10x=10: y=6(10)+4=60+4=64y = 6(10) + 4 = 60 + 4 = 64 (Matches)
    The rule is y=6x+4\boxed{y = 6x + 4}.

    QUESTION 9 [ALGEBRAIC EXPRESSIONS]

    9.1. Given the expression: x2−6x−8x^2 - 6x - 8

    9.1.1.

  10. Identify the terms in the expression.

    Terms are separated by addition or subtraction signs.
    The terms are x2x^2, −6x-6x, and −8-8.
    There are 3\boxed{3} terms in the expression.

    9.1.2.

  11. Identify the term containing xx.

    This is −6x-6x.

  12. The coefficient of xx is the numerical factor multiplying xx.

    The coefficient of xx is −6\boxed{-6}.

    9.1.3.

  13. Identify the term that does not contain any variables.

    The constant term is −8\boxed{-8}.

    9.1.4.

  14. Identify the power of the variable in each term.

    For x2x^2, the power is 2.
    For −6x-6x, the power is 1 (since x=x1x = x^1).
    For −8-8, the power is 0 (since x0=1x^0 = 1).

  15. The degree of the expression is the highest power of the variable in any term.

    The highest power is 2.
    The degree of the expression is 2\boxed{2}.

    9.1.5.

  16. Substitute x=−3x = -3 into the expression x2−6x−8x^2 - 6x - 8.

    (−3)2−6(−3)−8(-3)^2 - 6(-3) - 8
  17. Calculate the square of −3-3.

    (−3)2=(−3)×(−3)=9(-3)^2 = (-3) \times (-3) = 9
  18. Perform the multiplication.

    −6(−3)=18-6(-3) = 18
  19. Substitute these values back into the expression and perform the additions/subtractions.

    9+18−89 + 18 - 8 27−827 - 8 1919

    The value of the expression if x=−3x = -3 is 19\boxed{19}.

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