### Functions and graphs Examples

• Let k be a constant. If $$y = 4{x^2} - 6x + (2k + 3) > 0$$ for any real number x, find the range of values of k.
• ∵ There is no intersection between the graph and the x-axis.
∴ $$\begin{array}{1}\Delta < {0^{}}\\{( - 6)^2} - 4(4)(2k + 3) < 0\\36 - 32k - 48 < 0_{}^{}\\32k > - 12\\k > - \frac{{12}}{{32}}\\k > - \frac{3}{8}\end{array}$$
∴ The range of values of k is $$\underline{\underline {k > - {\rm{ }}\frac{3}{8}}}$$.

• It is given that the graph of the function $$y = 3{x^2} + px + q$$ cuts the x-axis at two points A(α, 0) and B(β, 0), where p and q are constants. The axis of symmetry of the graph cuts the x-axis at N(3, 0).
(a) Express $$\alpha + \beta$$ in terms of p. Hence find the value of p.
(b) If $$\alpha - \beta = 8$$, find the value of q.
(c) Find the values of α and β.
• (a) When $$y = 0$$,
$$3{x^2} + px + q = 0$$
$$\alpha + \beta =$$ Sum of roots $$= \underline{\underline { - {\rm{ }}\frac{p}{3}}}$$
∵ N is the mid-point of A and B.
$$\begin{array}{1}3 = \frac{{\alpha + \beta }}{2}\\∴\alpha + \beta = 6\;.............................\;(1)\\ - {\rm{ }}\frac{p}{3} = 6\\p = \underline{\underline { - {\rm{ }}18}} \end{array}$$
(b) α β = Product of roots $$= \frac{q}{3}$$
$$\begin{array}{1}\alpha - \beta = 8\;........\;(2)\\{(\alpha - \beta )^2} = 64\\{\alpha ^2} - 2\alpha \beta + {\beta ^2} = 64\\({\alpha ^2} + 2\alpha \beta + {\beta ^2}) - 4\alpha \beta = 64\\{(\alpha + \beta )^2} - 4\alpha \beta = 64\\{6^2} - 4(\frac{q}{3}) = 64\\\frac{{4q}}{3} = 36 - 64\\q = \underline{\underline { - {\rm{ }}21}} \end{array}$$
(c) (1) + (2), $$\begin{array}{1}(\alpha + \beta ) + (\alpha - \beta ) = 6 + 8\\2\alpha = 14\\\alpha = \underline{\underline 7} \end{array}$$
Substitute $$\alpha = 7$$ into (1)，
$$\begin{array}{1}7 + \beta = 6\\\beta = \underline{\underline { - {\rm{ }}1}} \end{array}$$

• A circus is going to perform in a venue which can hold 10 000 people. If the price of a ticket is $150, all tickets are expected to be sold. If the price of each ticket increases by$1, 40 less tickets will be sold. It is known that the total expenditure of the circus performance is $650 000. (a) Let$x be the price of each ticket and n be the number of tickets sold. Express n in terms of x.
(b) Let $y be the profit of the performance. Express y in terms of x. (c) How much should the ticket be sold to obtain the maximum profit? (d) To stimulate the sales of the tickets, a gift is given for every purchase of a ticket. As a result, 15% more tickets are sold than expected. It is known that the cost of each gift is$20. From the result of (c), find the profit obtained from the circus performance.
• (a) When the price of each ticket increases by $$\ (x - 150)$$,
$$40(x - 150)$$ less tickets will be sold.
$$\begin{array}{1}∴ n = 10{\rm{ }}000 - 40(x - 150)\\ = 10{\rm{ }}000 - 40x + 6{\rm{ }}000\\ = \underline{\underline {16{\rm{ }}000 - 40x}} \end{array}$$

$$\begin{array}{1}(b) y = x(16{\rm{ }}000 - 40x) - 650{\rm{ }}000\\ = \underline{\underline { - {\rm{ }}40{x^2} + 16{\rm{ }}000x - 650{\rm{ }}000}} \end{array}$$

(c) When y attains its maximum value,
$$\begin{array}{1}x = - \frac{{16{\rm{ }}000}}{{2( - {\rm{ }}40)}}\\ = 200\end{array}$$
∴ The ticket should be sold at \$200 to obtain the maximum profit.

(d) Expected number of tickets sold $$\begin{array}{l} = 16{\rm{ }}000 - 40(200)\\ = 8{\rm{ }}000\end{array}$$
Actual number of tickets sold $$\begin{array}{l} = 8{\rm{ }}000 \times (1 + 15\% )\\ = 9{\rm{ }}200\end{array}$$
∴ Profit obtained from the circus performance $$\begin{array}{l} = \ (9{\rm{ }}200 \times 200 - 650{\rm{ }}000 - 9{\rm{ }}200 \times 20)\\ = \underline{\underline {\ 1{\rm{ }}006{\rm{ }}000}} \end{array}$$