**3.7 Quadratic Functions, SPM Practice (Long Questions)**

Question 1:

Question 1:

Without drawing graph or using method of differentiation, find the maximum or minimum value of the function

*y*= 2 + 4*x*– 3*x*^{2}. Hence, find the equation of the axis of symmetry of the graph.

*Solution:*By completing the square for the function in the form of

*y*=*a*(*x*+*p*)^{2}+*q*to find the maximum or minimum value of the function.*y*= 2 + 4

*x*– 3

*x*

^{2}

*y*= – 3

*x*

^{2 }+ 4

*x*+ 2 ← (in general form)

$\begin{array}{l}y=-3\left[{x}^{2}-\frac{4}{3}x-\frac{2}{3}\right]\\ y=-3\left[{x}^{2}-\frac{4}{3}x+{\left(-\frac{4}{3}\times \frac{1}{2}\right)}^{2}-{\left(-\frac{4}{3}\times \frac{1}{2}\right)}^{2}-\frac{2}{3}\right]\\ y=-3\left[{\left(x-\frac{2}{3}\right)}^{2}-{\left(-\frac{2}{3}\right)}^{2}-\frac{2}{3}\right]\end{array}$

$\begin{array}{l}y=-3\left[{\left(x-\frac{2}{3}\right)}^{2}-\frac{4}{9}-\frac{6}{9}\right]\\ y=-3\left[{\left(x-\frac{2}{3}\right)}^{2}-\frac{10}{9}\right]\\ y=-3{\left(x-\frac{2}{3}\right)}^{2}+\frac{10}{3}\leftarrow \overline{)\text{intheformof}a{(x+p)}^{2}+q}\end{array}$

Since

*a*= –3 < 0,Therefore, the function

*y*has a maximum value of $\frac{10}{3}.$
$\begin{array}{l}x-\frac{2}{3}=0\\ x=\frac{2}{3}\end{array}$

Equation of the axis of symmetry of the graph is
$x=\frac{2}{3}.$
The diagram above shows the graph of a quadratic function

*y*=*f*(*x*). The straight line*y*= –4 is tangen to the curve*y*=*f*(*x*).(a) Write the equation of the axis of symmetry of the function

*f*(*x*).(b) Express

*f*(*x*) in the form of (*x*+*p*)^{2}+*q*, where*p*and*q*are constant.(c) Find the range of values of

*x*so that(i)

*f*(*x*) < 0, (ii)*f*(*x*) ≥ 0.

*Solution*

*:***(a)**

*x-*coordinate of the minimum point is the midpoint of (–2, 0) and (6, 0)

=
$=\frac{-2+6}{2}=2$

Therefore, equation of the axis of symmetry of the function

*f*(*x*) is*x*= 2.**(b)**

Substitute

*x*= 2 into*x*+*p*= 0,2 +

*p*= 0*p*= –2

and

*q*= –4 (the smallest value of*f*(*x*))Therefore,

*f*(*x*) = (*x*+*p*)^{2}+*q**f*(

*x*) = (

*x*– 2)

^{2}– 4

**(c)(i)**From the graph, for

*f*(

*x*) < 0, range of values of

*x*are –2 <

*x*< 6 ← (below

*x*-axis).

**(c)(ii)**From the graph, for

*f*(

*x*) ≥ 0, range of values of

*x*are

*x*≤ –2 atau

*x*≥ 6 ← (above

*x*-axis).