SPM 2022 Add Maths Paper 1, Question 14Quadratic Functions

Solved on video in English and Bahasa Melayu · Form 4, Quadratic Functions · 7:48 video

The question

(a)

14 Diagram 10 shows the curve of a quadratic function f(x) = 2x² + hx − 2k + 5, such that h and k are constants. The curve has a minimum point at T. From the diagram the curve cuts the x-axis at x = −2 and has axis of symmetry x = −1/4 (T lies below the x-axis). (a) Determine the range of values of x for f(x) > 0.

(b)

14 (b) Prove that k > (40 − h²/16.

(c)

14 (c) Using the method of completing the square, find the minimum value of the function f(x) = 2x² + hx − 2k + 5 in terms of k. [Note: from the axis of symmetry x = −1/4, h = 1.]

SPM 2022 Add Maths Paper 1, Question 14: the question as drawn in Eduly's video7:48

Part (a)

Step 1

The axis of symmetry x = −1/4 is the midpoint of the two roots. One root is x = −2; let the other be r.

Why this step

The axis of symmetry is exactly halfway between the roots, so one root plus the axis gives the other. No need to find h and k first.

4 more steps, each with the reason for it, in the full solution

Final answer

x < −2 or x > 3/2

Where students lose marks

Watch out: for f(x) > 0 on an upward parabola, take the regions outside the roots, not the interval between them.

Part (b)

6 more steps, each with the reason for it, in the full solution

Final answer

Proved

Where students lose marks

Watch out: two distinct real roots means b² − 4ac > 0. Using b² − 4ac < 0 gives the wrong inequality direction.

Part (c)

6 more steps, each with the reason for it, in the full solution

Final answer

Minimum value = −2k + 39/8

Where students lose marks

Watch out: factor 2 from the x-terms only and show every line. Dividing the whole function by 2 changes it and gives a wrong minimum.

Watch every step of this question solved on video (7:48), with the reason behind each step and a box that checks your own answer.

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