3.102: Absolute-value quadratic: stationary point and sketch

0606/12/F/M/21 — Question 4 · 9 marks

(a)
Show that 2x2+5x−32x^{2} + 5x - 3 can be written in the form a(x+b)2+ca\left(x + b\right)^{2} + c, where aa, bb and cc are constants.
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(b)
Hence write down the coordinates of the stationary point on the curve with equation y=∣2x2+5x−3∣y = \left\lvert 2x^{2} + 5x - 3\right\rvert.
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(c)
On the axes, sketch the graph of y=∣2x2+5x−3∣y = \left\lvert 2x^{2} + 5x - 3\right\rvert, stating the coordinates of the intercepts with the axes.
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(d)
Write down the value of kk for which the equation ∣2x2+5x−3∣=k\left\lvert 2x^{2} + 5x - 3\right\rvert = k has exactly 33 distinct solutions.
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