Functions and graphs Examples

Graph

Theory

ExamplesLet 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.

Solutions∵ There is no intersection between the graph and the xaxis.
∴ \(\begin{array}{1}\Delta < {0^{}}\\{(  6)^2}  4(4)(2k + 3) < 0\\36  32k  48 < 0_{}^{}\\32k >  12\\k >  \frac{{12}}{{32}}\\k >  \frac{3}{8}\end{array}\)
∴ The range of values of k is \(\underline{\underline {k >  {\rm{ }}\frac{3}{8}}} \).

Graph

Theory

ExamplesIt is given that the graph of the function \(y = 3{x^2} + px + q\) cuts the xaxis 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 xaxis 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. 
Solutions(a) When \(y = 0\),
\(3{x^2} + px + q = 0\)
\(\alpha + \beta = \) Sum of roots \( = \underline{\underline {  {\rm{ }}\frac{p}{3}}} \)
∵ N is the midpoint 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)，
\(\begin{array}{1}7 + \beta = 6\\\beta = \underline{\underline {  {\rm{ }}1}} \end{array}\)

Graph

Theory

ExamplesA 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. 
Solutions(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 \(\begin{array}{l} = \$ (9{\rm{ }}200 \times 200  650{\rm{ }}000  9{\rm{ }}200 \times 20)\\ = \underline{\underline {\$ 1{\rm{ }}006{\rm{ }}000}} \end{array}\)