NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
← Back to practice

Exploring functions and relations

Wide tables scroll horizontally. For detailed working, you may prefer a larger screen.

Required skills (click to reveal)

A craft designer is creating a symmetrical wooden inlay using the quadratic function f(x) = x^2 - 61x + 30 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.

Press Start when ready. Time includes time spent in another tab. Submit your answer before opening a help panel for a timed rank.

00:00
0 wordsOnly your numerical final answer or selected MCQ option is checked. Scratch working is not graded.

Report a problem or suggest an improvement

My notes — optional, not graded 0 words

Notes and scratch working stay in this browser and are linked to this question version. They are not graded or included in feedback. On shared devices, other people using this browser may see them.

For this question only. Your full notebook is on the dashboard.

Notes save in this browser. Download a copy to keep them elsewhere.

Report a problem or suggest an improvement

Your observations help improve this question. You can report an issue or recommend a better explanation, visual or experience.

0 words

The report includes your message and question/version details. Saved profile details, answers, notes and attempt history are not automatically attached. Leave out names and contact details, and check screenshots for personal information and embedded metadata.

Attachments may contain personal information. Review them before sharing. Logs collect only event codes and timestamps after you enable them. Download the ZIP to keep a copy, then attach it to an email if you choose.

Formula and explanation (click to reveal)

Opening this before your first submission makes the attempt practice only.

Formula

Explanation

Worked solution and answer (click to reveal)

Opening this before your first submission makes the attempt practice only.

Worked solution

  1. The quadratic function is given as f(x) = x^2 - 61x + 30. This is in standard form, f(x) = ax^2 + bx + c, where a = 1, b = -61, and c = 30.
  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 the values of a and b into the vertex formula: x = -(-61) / (2 * 1).
  4. Simplify the numerator: -(-61) = 61.
  5. Simplify the denominator: 2 * 1 = 2.
  6. Divide: 61 / 2 = 30.5.
  7. The x-coordinate of the vertex is 30.5. This is the point where the parabola reaches its maximum or minimum. Since a = 1 > 0, the parabola opens upwards, so the vertex is a minimum point.
  8. The question asks for the x-coordinate of the vertex, which is 30.5. This is an exact value, not an approximation.
  9. Check: substitute x = 30.5 into the vertex formula to confirm: x = -(-61)/(2*1) = 61/2 = 30.5. The calculation is consistent.
  10. Verify using symmetry: the roots of f(x) = 0 are symmetric about x = 30.5. The sum of the roots is 61 (from -b/a), so the midpoint is 61/2 = 30.5. This confirms the vertex lies at x = 30.5.
  11. The vertex x-coordinate is exactly 30.5, which is the point of symmetry and the minimum of the function.
  12. The exact x-coordinate of the vertex is 30.5.

Reference answer: 30.5

Alternative methods (click to reveal)

Shortcuts are shown only when applicable. Check the conditions before using one.

Earlier attempts for this question

#Date / timeYour answerResultSecondsHelp revealedRank
History stays in this browser when storage is available. Export it to your shared Drive folder and import it on another computer.
Worksheet details

Naming pattern: Topic_Grade_Complexity_YYYYMMDD_HH24MISS.html

Validation compares your answer with the generated reference. The model's reference can be wrong. Time ranks use an estimated student solving time, not a measured benchmark or a comparison with other students. An incorrect answer receives no speed rank.

NavGanitMate privacy & saved work

Saved in this browser. Your profile preferences, answers, practice history and notes are stored here when browser storage is available. They are not automatically uploaded to our servers. People sharing this browser may see saved work; changing the learner name does not create a separate private account. A nickname is enough.

Keep copies of important work. Clearing browser data, private browsing or changing browser or device can mean losing saved work. Download notes and export history separately. Downloads are copies of the selected content, not a complete account backup; not every format can be imported to restore your work. Keep these files somewhere private.

Visiting the website. Cloudflare helps deliver and protect this site and processes connection information such as IP addresses and requested pages. Server error logs may also contain request details. Security features may use cookies. This is separate from your browser-saved learning records. Cloudflare privacy information.

A parent-connection reminder uses a local list of up to five completed question IDs for this browser profile and a tab-session dismissal marker. Reminder activity is not uploaded and practice remains available.

Sharing feedback is your choice. A report includes your message and question/version details. Your saved profile, answers and notes are not automatically attached. Files you select can reveal personal details or contain embedded metadata: check and redact them before sharing. Diagnostic event codes and timestamps are included only after you enable logging. Sending email shares your message and attachments with the recipient and email providers. A download alone sends nothing to us. Website feedback delivery may be unavailable; only a confirmed receipt indicates it was received.