Question

(i)Express in the form $$4x^{2}-12x$$ in the form $$\left ( 2x+a \right )^{2}+b$$

(ii)Hence ,or otherwise,find the set of values of x satisfying $$4x^{2}-12x> 7$$

(i)(\left ( 2x-2 \right )^{2}-9\)

(ii)$$2x-3> 4$$    $$2x-3< – 4$$

$$x> 3\tfrac{1}{2}$$ (or)  $$x< -\frac{1}{2}$$

Allow $$-\frac{1}{2}> x>3\tfrac{1}{2}$$

OR

$$4x^{2}-12x-7\rightarrow \left ( 2x-7 \right )\left ( 2x+1 \right )$$

$$x> 3\tfrac{1}{2}$$  (or) $$< -\frac{1}{2}$$

Allow $$-\frac{1}{2}> x>3\tfrac{1}{2}$$

Question

1(i) Express $$x^{2}+6x+2$$ in the form $$\left ( x+a \right )^{2}+b$$,where a and b are constants.

(ii)Henc3,or otherwise, find the set of values of x for which $$x^{2}+6x+2> 9$$.

(i)$$\left ( x+3 \right )^{2}-7$$

(ii)1,-7 seen

$$x> 1,x< -7$$

Question

(i) Express $$-x^{2}+6x-5$$ in the form $$a(x+b)^{2}+c$$, where a, b and c are constants.

The function $$f:x\rightarrow -x^{2}+6x-5$$ defined for $$x\geq m$$ ,where m is constant.

(ii) State the smallest value of m for which f is one-one.
(iii) For the case where m = 5, find an expression for $$f^{-1}(x)$$ and state the domain of $$f^{-1}$$.

(i)$$-1(x-3)^{2}+4$$

(ii)Smallest (m)-is 3

(iii)$$(x-3)^{2}=4-y$$

Correct order of operations

$$f^{-1}(x)=3+\sqrt{4-x}$$

Domain is$$x\leq 0$$

.

Question

(i) Express $$x^{2}-2x-15$$  in the form $$(x+a)^{2}+b$$.

The function f is defined for p ≤ x ≤ q, where p and q are positive constants, by f : x → $$x^{2}$$ − 2x − 15.
The range of f is given by c ≤ fx ≤ d, where c and d are constants.
(ii) State the smallest possible value of c.
For the case where c = 9 and d = 65,
(iii) find p and q,
(iv) find an expression for $$f^{-1}(x)$$.

(i)$$(x-1)^{2}-16$$

(ii)-16

(iii)$$9\leq (x-1)^{2}-16\leq 65$$ OR $$x^{2}-2x-15=9\rightarrow 6,-4$$

$$25\leq (x-1)^{2}\leq 81$$ $$x^{2}-2x-15=65\rightarrow 10,-8$$

$$5\leq x-1\leq 9$$             p=6

$$6\leq x\leq 10$$        q=10

(iv)$$x=\left ( y-1 \right )^{2}-16$$      (interchange x/y)

$$y-1=\pm \sqrt{x+16}$$

f^{-1}\left ( x \right )=1+\sqrt{x+16}

Question

A curve for which $$\frac{\mathrm{d} y}{\mathrm{d} x}=7-x^{2}-6x$$   pases tghrough the p[oint (3,-10)

(i)Find the equation of the curve.

(ii)Express $$7-x^{2}-6x$$ in the form $$a-(x+b)^{2}$$,where a and b are constants.

(iii)Find the set of values of x for which the gradient of the curve is positive.

(i)$$y=7x-\frac{x^{3}}{3}-\frac{6x^{2}}{2}+c$$

Uses (3,-10) →c=5

(ii)$$7-x^{2}-6x=16-(x+3)^{2}$$

(iii)$$16-\left ( x+3 \right )^{2}> 0\rightarrow \left ( x+3 \right )^{2}< 16$$,and solve

End-points x=1 or -7

$$\rightarrow -7< x< 1$$

### Question

The graph of y = f(x) is transformed to the graph of $$y = 1 + f(\frac{1}{2}x)$$.
Describe fully the two single transformations which have been combined to give the resulting transformation.

Ans:

[Stretch] [factor 2, x direction (or y-axis invariant)]
[Translation or Shift][1 unit in y direction] or
[Translation/Shift] $$[\binom{0}{1}]$$

Question

The function f is defined by $$f(x)=-2x^{2}+12x-3 for \varepsilon R$$

(i) Express $$-2x^{2}+12x-3 in the from 2(x+a)^{a}+b,$$where a and b are constants.

(ii) State the greatest value of fx.

The function g is defined by g(x) = 2x + 5 for x ∈ >.

(iii) Find the values of x for which gf(x )+ 1

(i) $$-2(x-3) ^{2}$$+15(a=-3,b=15)
(iii) gf(x) =$$(-2x^{2}+12x-3)+5=-4x^{2}+24x-6+5 gf(x)+1=0\rightarrow -4x^{2}+24x$$=0 x=0 or 6