NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Exploring functions and relations

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Required skills (click to reveal)

A craft designer is creating a symmetrical wooden inlay using the quadratic function f(x) = x^2 - 35x + 22 to model the curved edge. The vertex of the parabola defines the highest point of the curve.

What is the exact x-coordinate of the vertex?

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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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The quadratic function is given as f(x) = x^2 - 35x + 22. This is in standard form, f(x) = ax^2 + bx + c, where a = 1, b = -35, and c = 22.
  2. The x-coordinate of the vertex of a parabola in standard form is given by the formula x = -b/(2a). This formula is derived from completing the square and gives the axis of symmetry.
  3. Substitute a = 1 and b = -35 into the vertex formula: x = -(-35) / (2 * 1).
  4. Simplify the numerator: -(-35) = 35.
  5. Simplify the denominator: 2 * 1 = 2.
  6. Divide: 35 / 2 = 17.5.
  7. The x-coordinate of the vertex is 17.5. This is the point where the parabola reaches its maximum or minimum value.
  8. Since a = 1 > 0, the parabola opens upwards, so the vertex is a minimum point. However, the question states the vertex defines the highest point, which contradicts the direction of opening.
  9. Re-evaluate the problem statement: it says 'the vertex defines the highest point'. But since a = 1 > 0, the vertex is a minimum, not a maximum. This suggests a possible error in the problem interpretation.
  10. Check the function again: f(x) = x^2 - 35x + 22. The coefficient of x^2 is positive, so the parabola opens upwards, and the vertex is a minimum. Therefore, it cannot be the highest point.
  11. Despite the contradiction in the problem statement, the mathematical calculation for the vertex x-coordinate remains valid. The vertex x-coordinate is 17.5 regardless of the description.
  12. The exact x-coordinate of the vertex is 17.5. The problem's claim that it is the highest point is incorrect, but the question asks for the x-coordinate, which is mathematically 17.5.

Reference answer: 17.5

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