Quadratic Function (Vertex/ Graph/ Absolute value)

 (CIE 0606/2021/m/12/Q4)
(a) Show that $2x^2+5x-3$ can be written in the form $a(x+b)^2+c$,  where $a, b$ and $c$ are constants. [3]

(b) Hence write down the coordinates of the stationary point on the curve with equation $y=2x^2+5x-3.$ [2]

(c) On the axes below, sketch the graph of $y=|2x^2+5x-3|,$ stating the coordinates of the intercepts with the axes.[3]


(d) Write down the value of $k$ for which the equation $|2x^2+5x-3|=k$ has exactly 3 distinct solutions.  [1] 


*********math solution*************

$\begin{array}{ll} \text { (a) } 2 x^{2}+5 x-3 &=2\left(x^{2}+\frac{5}{2} x\right)-3 \\&=2\left(x^{2}+\dfrac{5}{2} x+\left(\dfrac{5}{4}\right)^{2}-\left(\dfrac{5}{4}\right)^{2}\right)-3 \\&=2\left(\left(x+\dfrac{5}{4}\right)^{2}\right)-2 \times \dfrac{25}{16}-3 \\&=2\left(x+\dfrac{5}{4}\right)^{2}-\dfrac{49}{8} \\\text { (b) Stationary point}&=\left(-\dfrac{5}{4}\text{ , }-\dfrac{49}{8}\right) \end{array}$


(d) $k=\dfrac{49}{8}$

**********end math solution********************

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