Read the Minimum Point from Vertex Form
In y=(x−h)²+k, the square is nonnegative and becomes zero at x=h.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Find the vertex of y=(x−2)²−3. Candidate answers: (A) (0,1) (B) (2,3) (C) (2,−3) (D) (−2,−3).
- Independent check inputs: Find the vertex of y=(x−3)²−4. Candidate answers: (A) (0,5) (B) (3,4) (C) (3,−4) (D) (−3,−4).
Learn the idea
Quadratic equations and graphs express the same underlying structure. A discriminant predicts real-root count, the quadratic formula calculates roots, and vertex form displays shifts, direction and stretching directly. In y=(x−h)²+k, the square is nonnegative and becomes zero at x=h. Therefore the smallest output is k, with vertex coordinates (h,k).
Here is the method on the supplied teaching case. In y=(x−h)²+k, the square is least at x=h. Here h=2 and k=−3, so the vertex is (2,−3).
The sign inside x−h is opposite the horizontal coordinate’s written sign. The vertex is not generally the y-intercept; the latter uses x=0 and can be found separately.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Find the vertex of y=(x−2)²−3. Candidate answers: (A) (0,1) (B) (2,3) (C) (2,−3) (D) (−2,−3). Solution: In y=(x−h)²+k, the square is least at x=h. Here h=2 and k=−3, so the vertex is (2,−3). The final answer is (2,−3). The sign inside x−h is opposite the horizontal coordinate’s written sign. The vertex is not generally the y-intercept; the latter uses x=0 and can be found separately.
Check your reasoning
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Helpful tips
- In y=(x−h)²+k, the square is nonnegative and becomes zero at x=h.
- The sign inside x−h is opposite the horizontal coordinate’s written sign.
See the answer and explanation
(3,−4)
In y=(x−h)²+k, the square is least at x=h. Here h=3 and k=−4, so the vertex is (3,−4).
Take it further
Changed-input task: Find the vertex of y=(x−9)²−10. Candidate answers: (A) (0,71) (B) (9,10) (C) (9,−10) (D) (−9,−10). Show the mathematical reason, not only the selected answer. Expected result: (9,−10). In y=(x−h)²+k, the square is least at x=h. Here h=9 and k=−10, so the vertex is (9,−10).
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