NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Graph transformations and parameters

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Required skills (click to reveal)

A community garden plot is designed using the graph of y = x^2. The design is transformed by a vertical stretch by a factor of 1/4, a horizontal translation 7 units to the right, and a vertical translation 11 units up.

What is the y-coordinate of the point on the transformed graph when x = 44? Express your answer as an exact value.

Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

Moving a parabola up three units. The solid parabola is y = x² + 3; the dashed parabola is y = x². Every output increases by 3, moving the vertex from (0,0) to (0,3).xy(0,3)y = x² + 3
The solid parabola is y = x² + 3; the dashed parabola is y = x². Every output increases by 3, moving the vertex from (0,0) to (0,3).
A straight-line relationship. Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.-2-2-1-111223344xy
Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The original function is y = x^2. The transformations applied are: a vertical stretch by factor 1/4, a horizontal translation 7 units right, and a vertical translation 11 units up.
  2. The general form of the transformed function is y = a(x - h)^2 + k, where a is the vertical stretch factor, h is the horizontal shift, and k is the vertical shift.
  3. Substitute the given values: a = 1/4, h = 7, k = 11. The transformed function becomes y = (1/4)(x - 7)^2 + 11.
  4. To find the y-coordinate when x = 44, substitute x = 44 into the transformed equation.
  5. Calculate (x - 7) = 44 - 7 = 37.
  6. Square the result: 37^2 = 1369.
  7. Multiply by the vertical stretch factor: (1/4) × 1369 = 1369/4.
  8. Add the vertical translation: 1369/4 + 11 = 1369/4 + 44/4 = (1369 + 44)/4 = 1413/4.
  9. Simplify the fraction: 1413 ÷ 4 = 353.25, but keep as exact fraction 1413/4.
  10. The y-coordinate is 1413/4. This is the exact value.
  11. Check: substitute x = 44 into y = (1/4)(x - 7)^2 + 11. (44 - 7)^2 = 37^2 = 1369. 1369 × 1/4 = 342.25. 342.25 + 11 = 353.25 = 1413/4. Correct.
  12. The y-coordinate when x = 44 is 1413/4.

Reference answer: 1413/4

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