New Discovering Mathematics 3A Textbook solutions
Ex 1.6
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Solutions
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(a)
\begin{align} y & = (x - 1)(x - 4) \\ y & = x^2 - 4x - x + 4 \\ y & = x^2 - 5x + 4 \phantom{0000000} [\text{Minimum curve } \cup] \\ \\ \text{Let } & x = 0, \\ y & = (0 - 1)(0 - 4) \\ y & = 4 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x - 1)(x - 4) \\ \\ x - 1 & = 0 \phantom{0} \text{ or } \phantom{0} x - 4 = 0 \\ x & = 1 \phantom{00000000} x = 4 \phantom{0000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {1 + 4 \over 2} \\ x & = 2.5 \\ \\ \text{Let } & x = 2.5, \\ y & = (2.5 - 1)(2.5 - 4) \\ y & = -2.25 \\ \\ \text{Turning point: } & (2.5, -2.25) \end{align}
(b)
\begin{align} y & = (x + 3)(x - 1) \\ y & = x^2 - x + 3x - 3 \\ y & = x^2 + 2x - 3 \phantom{0000000} [\text{Minimum curve } \cup] \\ \\ \text{Let } & x = 0, \\ y & = (0 + 3)(0 - 1) \\ y & = - 3 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x + 3)(x - 1) \\ \\ x + 3 & = 0 \phantom{0} \text{ or } \phantom{0} x - 1 = 0 \\ x & = -3 \phantom{000000(} x = 1 \phantom{00000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {-3 + 1 \over 2} \\ x & = -1 \\ \\ \text{Let } & x = -1, \\ y & = (-1 + 3)(-1 - 1) \\ y & = -4 \\ \\ \text{Turning point: } & (-1, -4) \end{align}
(c)
\begin{align} y & = -(x + 2)(x + 3) \\ y & = -(x^2 + 3x + 2x + 6) \\ y & = -(x^2 + 5x + 6) \\ y & = -x^2 - 5x - 6 \phantom{0000000} [\text{Maximum curve } \cap] \\ \\ \text{Let } & x = 0, \\ y & = -(0 + 2)(0 + 3) \\ y & = -6 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = -(x + 2)(x + 3) \\ \\ x + 2 & = 0 \phantom{0} \text{ or } \phantom{0} x + 3 = 0 \\ x & = -2 \phantom{000000(} x = -3 \phantom{0000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {-2 + (-3) \over 2} \\ x & = -2.5 \\ \\ \text{Let } & x = -2.5, \\ y & = -(-2.5 + 2)(-2.5 + 3) \\ y & = 0.25 \\ \\ \text{Turning point: } & (-2.5, 0.25) \end{align}
(d)
\begin{align} y & = -x(x + 5) \\ y & = - x^2 - 5x \phantom{0000000} [\text{Maximum curve } \cap] \\ \\ \text{Let } & x = 0, \\ y & = -(0)(0 + 5) \\ y & = 0 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = -x(x + 5) \\ \\ -x & = 0 \phantom{0} \text{ or } \phantom{0} x + 5 = 0 \\ x & = 0 \phantom{00000000} x = -5 \phantom{0000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {0 + (-5) \over 2} \\ x & = -2.5 \\ \\ \text{Let } & x = -2.5, \\ y & = -(-2.5)(-2.5 + 5) \\ y & = 6.25 \\ \\ \text{Turning point: } & (-2.5, 6.25) \end{align}
(a)
\begin{align} y & = x^2 + 2 \phantom{0000000} [\text{Minimum curve } \cup] \\ y & = (x - 0)^2 + 2 \\ \\ \text{Turning point: } & (0, 2) \\ \\ \text{Let } & x = 0, \\ y & = (0)^2 + 2 \\ y & = 2 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(b)
\begin{align} y & = (x + 1)^2 \phantom{0000000} [\text{Minimum curve } \cup] \\ y & = (x + 1)^2 + 0 \\ \\ \text{Turning point: } & (-1, 0) \\ \\ \text{Let } & x = 0, \\ y & = (0 + 1)^2 \\ y & = 1 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(c)
\begin{align} y & = (x - 3)^2 + 4 \phantom{0000000} [\text{Minimum curve } \cup] \\ \\ \text{Turning point: } & (3, 4) \\ \\ \text{Let } & x = 0, \\ y & = (0 - 3)^2 + 4 \\ y & = 13 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(d)
\begin{align} y & = (x + 2)^2 - 5 \phantom{0000000} [\text{Minimum curve } \cup] \\ \\ \text{Turning point: } & (-2, -5) \\ \\ \text{Let } & x = 0, \\ y & = (0 + 2)^2 - 5 \\ y & = -1 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(e)
\begin{align} y & = (x + 3)^2 + 2 \phantom{0000000} [\text{Minimum curve } \cup] \\ \\ \text{Turning point: } & (-3, 2) \\ \\ \text{Let } & x = 0, \\ y & = (0 + 3)^2 + 2 \\ y & = 11 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(f)
\begin{align} y & = (x - 4)^2 \phantom{0000000} [\text{Minimum curve } \cup] \\ y & = (x - 4)^2 + 0 \\ \\ \text{Turning point: } & (4, 0) \\ \\ \text{Let } & x = 0, \\ y & = (0 - 4)^2 \\ y & = 16 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(a)
\begin{align} y & = -x^2 - 3 \phantom{0000000} [\text{Maximum curve } \cap] \\ y & = -(x - 0)^2 - 3 \\ \\ \text{Turning point: } & (0, -3) \\ \\ \text{Line of symmetry, } & x = 0 \\ \\ \text{Let } & x = 0, \\ y & = -(0)^2 - 3 \\ y & = -3 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(b)
\begin{align} y & = -(x - 4)^2 \phantom{0000000} [\text{Maximum curve } \cap] \\ y & = -(x - 4)^2 + 0 \\ \\ \text{Turning point: } & (4, 0) \\ \\ \text{Line of symmetry, } & x = 4 \\ \\ \text{Let } & x = 0, \\ y & = -(0 - 4)^2 \\ y & = -16 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(c)
\begin{align} y & = -(x + 1)^2 - 2 \phantom{0000000} [\text{Maximum curve } \cap] \\ \\ \text{Turning point: } & (-1, -2) \\ \\ \text{Line of symmetry, } x & = -1 \\ \\ \text{Let } & x = 0, \\ y & = -(0 + 1)^2 - 2 \\ y & = -3 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(d)
\begin{align} y & = -(x - 2)^2 + 3 \phantom{0000000} [\text{Maximum curve } \cap] \\ \\ \text{Turning point: } & (2, 3) \\ \\ \text{Line of symmetry, } & x = 2 \\ \\ \text{Let } & x = 0, \\ y & = - (0 - 2)^2 + 3 \\ y & = -1 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(e)
\begin{align} y & = -(x + 2)^2 \phantom{0000000} [\text{Maximum curve } \cap] \\ y & = -(x + 2)^2 + 0 \\ \\ \text{Turning point: } & (-2, 0) \\ \\ \text{Line of symmetry, } & x = -2 \\ \\ \text{Let } & x = 0, \\ y & = -(0 + 2)^2 \\ y & = -4 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(f)
\begin{align} y & = -(x + 3)^2 + 1 \phantom{0000000} [\text{Maximum curve } \cap] \\ \\ \text{Turning point: } & (-3, 1) \\ \\ \text{Line of symmetry, } & x = -3 \\ \\ \text{Let } & x = 0, \\ y & = -(0 + 3)^2 + 1 \\ y & = -8 \phantom{0000000000000000} [y \text{-intercept}] \end{align}
(a)
\begin{align} y & = 1 - x^2 \\ y & = -x^2 + 1 \phantom{0000000} [\text{Maximum curve } \cap] \\ y & = -(x - 0)^2 + 1 \\ \\ \text{Turning point: } & (0, 1) \\ \\ \text{Let } & x = 0, \\ y & = 1 - 0^2 \\ y & = 1 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = 1 - x^2 \\ x^2 & = 1 \\ x & = \pm \sqrt{1} \\ x & = \pm 1 \phantom{000000000000000} [x \text{-intercepts}] \end{align}
(b)
$$ x = 0 $$
(a)
\begin{align} y & = x^2 + 3x - 4 \phantom{0000000} [\text{Minimum curve } \cup] \\ y & = (x + 4)(x - 1) \\ \\ \text{Let } & x = 0, \\ y & = (0 + 4)(0 - 1) \\ y & = -4 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x + 4)(x - 1) \\ \\ x + 4 & = 0 \phantom{0} \text{ or } \phantom{0} x - 1 = 0 \\ x & = -4 \phantom{000000(} x = 1 \phantom{00000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {-4 + 1 \over 2} \\ x & = -1.5 \\ \\ \text{Let } & x = -1.5, \\ y & = (-1.5 + 4)(-1.5 - 1) \\ y & = -6.25 \\ \\ \text{Turning point: } & (-1.5, -6.25) \end{align}
(b)
\begin{align} y & = -x^2 + 3x - 2 \phantom{0000000} [\text{Maximum curve } \cap] \\ y & = -(x^2 - 3x + 2) \\ y & = -(x - 1)(x - 2) \\ \\ \text{Let } & x = 0, \\ y & = -(0 - 1)(0 - 2) \\ y & = -2 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = -(x - 1)(x - 2) \\ \\ x - 1 & = 0 \phantom{0} \text{ or } \phantom{0} x - 2 = 0 \\ x & = 1 \phantom{00000000} x = 2 \phantom{00000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {1 + 2 \over 2} \\ x & = 1.5 \\ \\ \text{Let } & x = 1.5, \\ y & = -(1.5 - 1)(1.5 - 2) \\ y & = 0.25 \\ \\ \text{Turning point: } & (1.5, 0.25) \end{align}
(c)
\begin{align} y & = -x^2 - 2x - 1 \phantom{0000000} [\text{Maximum curve } \cap] \\ y & = -(x^2 + 2x + 1) \\ y & = -(x + 1)(x + 1) \\ y & = -(x + 1)^2 \\ \\ \text{Let } & x = 0, \\ y & = -(0 + 1)^2 \\ y & = -1 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = -(x + 1)^2 \\ 0 & = x + 1 \\ -1 & = x \phantom{000000000000} [x \text{-intercept, also turning point}] \\ \\ \text{Turning point: } & (-1, 0) \end{align}
(d)
\begin{align} y & = x^2 - 4 \phantom{000000000} [\text{Minimum curve } \cup] \\ y & = x^2 - 2^2 \\ y & = (x + 2)(x - 2) \phantom{000} [a^2 - b^2 = (a + b)(a - b)] \\ \\ \text{Let } & x = 0, \\ y & = (0 + 2)(0 - 2) \\ y & = -4 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x + 2)(x - 2) \\ \\ x + 2 & = 0 \phantom{0} \text{ or } \phantom{0} x - 2 = 0 \\ x & = -2 \phantom{000000.} x = 2 \phantom{00000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {-2 + 2 \over 2} \\ x & = 0 \\ \\ \text{Turning point: } & (0, -4) \phantom{000000000} [\text{Same as } y \text{-intercept}] \end{align}
(e)
\begin{align} y & = -6 - 5x + x^2 \\ y & = x^2 - 5x - 6 \phantom{000000000} [\text{Minimum curve } \cup] \\ y & = (x + 1)(x - 6) \\ \\ \text{Let } & x = 0, \\ y & = (0 + 1)(0 - 6) \\ y & = -6 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x + 1)(x - 6) \\ \\ x + 1 & = 0 \phantom{0} \text{ or } \phantom{0} x - 6 = 0 \\ x & = -1 \phantom{000000.} x = 6 \phantom{00000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {-1 + 6 \over 2} \\ x & = 2.5 \\ \\ \text{Let } & x = 2.5, \\ y & = (2.5 + 1)(2.5 - 6) \\ y & = -12.25 \\ \\ \text{Turning point: } & (2.5, -12.25) \end{align}
(f)
\begin{align} y & = 6x - x^2 - 9 \\ y & = - x^2 + 6x - 9 \phantom{000000000} [\text{Maximum curve } \cap] \\ y & = - (x^2 - 6x + 9) \\ y & = - (x - 3)(x - 3) \\ y & = -(x - 3)^2 \\ \\ \text{Let } & x = 0, \\ y & = - (0 - 3)^2 \\ y & = - 9 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = - (x - 3)^2 \\ 0 & = (x - 3)^2 \\ 0 & = x - 3 \\ 3 & = x \phantom{000000000000000000} [x \text{-intercept & turning point}] \\ \\ \text{Line of symmetry, } x & = 3 \\ \\ \text{Turning point: } & (3, 0) \end{align}
(a)
\begin{align} y & = (x - p)(x - q) \phantom{000000} [\text{Minimum curve } \cup] \\ \\ \text{When } & y = 0, x = -1 \text{ or } x = 2 \\ \\ x & = -1 \phantom{00} \text{ or } \phantom{0000} x = 2 \\ x + 1 & = 0 \phantom{000000.} x - 2 = 0 \\ \\ & (x + 1)(x - 2) = 0 \\ \\ \therefore y & = (x + 1)(x - 2) \end{align}
(b)
\begin{align} \text{Line of sym} & \text{metry, } x = -2 \\ \\ \text{Let } a \text{ represent} & \text{ the other } x \text{-intercept} \\ \\ -2 & = {-6 + a \over 2} \\ {-4 \over 2} & = {-6 + a \over 2} \\ -4 & = -6 + a \\ -4 + 6 & = a \\ 2 & = a \\ \\ \\ y & = -(x - p)(x - q) \phantom{000000} [\text{Maximum curve } \cup] \\ \\ \text{When } & y = 0, x = -6 \text{ or } x = 2 \\ \\ x & = -6 \phantom{00} \text{ or } \phantom{0000} x = 2 \\ x + 6 & = 0 \phantom{000000.} x - 2 = 0 \\ \\ & (x + 6)(x - 2) = 0 \\ \\ \therefore y & = -(x + 6)(x - 2) \end{align}
(a)
\begin{align} y & = - (x - h)^2 + k \phantom{000000} [\text{Maximum curve}] \\ \\ \text{Turning } & \text{point: } (5, -2) \\ \\ y & = -(x - 5)^2 - 2 \end{align}
(b)
\begin{align} y & = (x - h)^2 + k \phantom{000000} [\text{Minimum curve}] \\ \\ \text{Let } a \text{ rep} & \text{resent the } y \text{-coordinate of the turning point} \\ \\ \text{Turning } & \text{point: } (-2, a) \\ \\ y & = (x + 2)^2 + a \\ \\ \text{Using } & (0, 5), \phantom{0000000000000} [y \text{-intercept}] \\ 5 & = (0 + 2)^2 + a \\ 5 & = 4 + a \\ 5 - 4 & = a \\ 1 & = a \\ \\ \therefore y & = (x + 2)^2 + 1 \end{align}
(a)
\begin{align} y & = x^2 + 2x + 3 \phantom{0000000000000000000} [\text{Minimum curve } \cup] \\ y & = x^2 + 2x + \left(2 \over 2\right)^2 - \left(2 \over 2\right)^2 + 3 \\ y & = x^2 + 2x + 1^2 - 1^2 + 3 \\ y & = (x + 1)^2 - 1 + 3 \\ y & = (x + 1)^2 + 2 \\ \\ \text{Turning } & \text{point: } (-1, 2) \\ \\ \text{Line of } & \text{symmetry, } x = -1 \\ \\ \text{Let } & x = 0, \\ y & = (0 + 1)^2 + 2 \\ y & = 3 \phantom{00000000000000} [y \text{-intercept}] \end{align}
(b)
\begin{align} y & = - x^2 + 8x - 5 \phantom{0000000000000000000} [\text{Maximum curve } \cap] \\ y & = -(x^2 - 8x) - 5 \\ y & = - \left[ x^2 - 8x + \left(8 \over 2\right)^2 - \left(8 \over 2\right)^2 \right] - 5 \\ y & = - ( x^2 - 8x + 4^2 - 4^2 ) - 5 \\ y & = - [ (x - 4)^2 - 16 ] - 5 \\ y & = - (x - 4)^2 + 16 - 5 \\ y & = - (x - 4)^2 + 11 \\ \\ \text{Turning } & \text{point: } (4, 11) \\ \\ \text{Line of } & \text{symmetry, } x = 4 \\ \\ \text{Let } & x = 0, \\ y & = -(0 - 4)^2 + 11 \\ y & = -5 \phantom{00000000000000} [y \text{-intercept}] \end{align}
(c)
\begin{align} y & = 5x - x^2 \\ y & = -x^2 + 5x \phantom{0000000000000000000} [\text{Maximum curve } \cap] \\ y & = -(x^2 - 5x) \\ y & = -\left[ x^2 - 5x + \left(5 \over 2\right)^2 - \left(5 \over 2\right)^2 \right] \\ y & = - (x^2 - 5x + 2.5^2 - 2.5^2) \\ y & = - [ (x - 2.5)^2 - 6.25) ] \\ y & = - (x - 2.5)^2 + 6.25 \\ \\ \text{Turning } & \text{point: } (2.5, 6.25) \\ \\ \text{Line of } & \text{symmetry, } x = 2.5 \\ \\ \text{Let } & x = 0, \\ y & = - (0 - 2.5)^2 + 6.25 \\ y & = 0 \phantom{0000000000000000000000000} [y \text{-intercept}] \end{align}
(d)
\begin{align} y & = x^2 - 7x + 6 \phantom{0000000000000000000} [\text{Minimum curve } \cup] \\ y & = x^2 - 7x + \left(7 \over 2\right)^2 - \left(7 \over 2\right)^2 + 6 \\ y & = x^2 - 7x + 3.5^2 - 3.5^2 + 6 \\ y & = (x - 3.5)^2 - 12.25 + 6 \\ y & = (x - 3.5)^2 - 6.25 \\ \\ \text{Turning } & \text{point: } (3.5, -6.25) \\ \\ \text{Line of } & \text{symmetry, } x = 3.5 \\ \\ \text{Let } & x = 0, \\ y & = (0 - 3.5)^2 - 6.25 \\ y & = 6 \phantom{0000000000000000000000000000} [y \text{-intercept}] \end{align}
(e)
\begin{align} y & = x^2 + 4x + 4 \phantom{0000000000000000000} [\text{Minimum curve } \cup] \\ y & = x^2 + 4x + \left(4 \over 2\right)^2 - \left(4 \over 2\right)^2 + 4 \\ y & = x^2 + 4x + 2^2 - 2^2 + 4 \\ y & = (x + 2)^2 - 4 + 4 \\ y & = (x + 2)^2 + 0 \\ \\ \text{Turning } & \text{point: } (-2, 0) \\ \\ \text{Line of } & \text{symmetry, } x = -2 \\ \\ \text{Let } & x = 0, \\ y & = (0 + 2)^2 + 0 \\ y & = 4 \phantom{0000000000000000000000000000} [y \text{-intercept}] \end{align}
(f)
\begin{align} y & = -x^2 - 10x - 25 \phantom{0000000000000000000} [\text{Maximum curve } \cap] \\ y & = - (x^2 + 10x) - 25 \\ y & = - \left[ x^2 + 10x + \left(10 \over 2\right)^2 - \left(10 \over 2\right)^2 \right] - 25 \\ y & = - ( x^2 + 10x + 5^2 - 5^2 ) - 25 \\ y & = - [ (x + 5)^2 - 25 ] - 25 \\ y & = - (x + 5)^2 + 25 - 25 \\ y & = - (x + 5)^2 \\ \\ \text{Turning } & \text{point: } (-5, 0) \\ \\ \text{Line of } & \text{symmetry, } x = -5 \\ \\ \text{Let } & x = 0, \\ y & = - (0 + 5)^2 \\ y & = 0 \phantom{0000000000000000000000000000} [y \text{-intercept}] \end{align}
(a)
\begin{align} y & = x^2 - 2x - 8 \\ \\ \text{Let } & x = 0, \\ y & = (0)^2 - 2(0) - 8 \\ y & = -8 \\ \\ \therefore & \phantom{.} C(0, -8) \end{align}
(b)
\begin{align} y & = x^2 - 2x - 8 \\ \\ \text{Let } & y = 0, \\ 0 & = x^2 - 2x - 8 \\ 0 & = (x + 2)(x - 4) \end{align} \begin{align} x + 2 & = 0 & & \text{ or } & x - 4 & =0 \\ x & = - 2 &&& x & = 4 \\ \\ \therefore & \phantom{.} A(-2, 0) &&& \therefore & \phantom{.} B(4, 0) \end{align}
(c)
\begin{align} \text{Line of symmetry, } x & = {-2 + 4 \over 2} \\ x & = 1 \\ \\ \text{Substitute } & x = 1 \text{ into eqn of curve,} \\ y & = (1)^2 - 2(1) - 8 \\ y & = -9 \\ \\ \text{Minimum } & \text{point: } (1, -9) \end{align}
(a)
\begin{align} y & = x^2 + 2x + k \\ \\ \text{Using } & P(-3, 0), \\ 0 & = (-3)^2 + 2(-3) + k \\ 0 & = 9 - 6 + k \\ 0 & = 3 + k \\ -3 & = k \end{align}
(b)
\begin{align} y & = x^2 + 2x + k \\ y & = x^2 + 2x - 3 \\ \\ \text{Let } & x = 0, \\ y & = (0)^2 + 2(0) - 3 \\ y & = -3 \\ \\ \therefore & \phantom{.} R(0, -3) \\ \\ \\ \text{Let } & y = 0, \\ 0 & = x^2 + 2x - 3 \\ 0 & = (x + 3)(x - 1) \end{align} \begin{align} x + 3 & = 0 & & \text{ or } & x - 1 & =0 \\ x & = - 3 &&& x & = 1 \\ \\ \therefore & \phantom{.} P(-3, 0) &&& \therefore & \phantom{.} Q(1, 0) \end{align}
(c)
\begin{align} \text{Line of symmetry, } x & = {-3 + 1 \over 2} \\ x & = -1 \end{align}
(a)
\begin{align} y & = -2x^2 + 48x - 160 \phantom{0000000000} [\text{Maximum curve } \cap] \\ y & = -2(x^2 - 24x) - 160 \\ y & = -2 \left[ x^2 - 24x + \left(24 \over 2\right)^2 - \left(24 \over 2\right)^2 \right] - 160 \\ y & = -2 ( x^2 - 24x + 12^2 - 12^2 ) - 160 \\ y & = -2 [ (x - 12)^2 - 144 ] - 160 \\ y & = -2 (x - 12)^2 + 288 - 160 \\ y & = -2 (x - 12)^2 + 128 \\ \\ \text{Turning } & \text{point: } (12, 128) \\ \\ \text{Let } & x = 0, \\ y & = -2(0 - 12)^2 + 128 \\ y & = -160 \phantom{000000000000000000000} [y \text{-intercept}] \end{align}
(b)
\begin{align} \text{From (a), } & y = -160 \\ \\ \text{Sam loses \$ 160 daily } & \text{if he doesn't make any vase} \end{align}
(c)(i)
\begin{align} \text{Turning point} & \text{: } (12, 128) \\ \\ \text{No. of vases }(x) & = 12 \end{align}
(c)(ii)
\begin{align} \text{Turning point} & \text{: } (12, 128) \\ \\ \text{Max. daily profit } (y) & = \$ 128 \end{align}
(a)
\begin{align}
\text{Let coordinates} & \text{ of } R \text{ be } (r, 0) \\
\\
\text{Line of symmetry, } x & = {-5 + r \over 2} \\
1 & = {-5 + r \over 2} \\
{2 \over 2} & = {-5 + r \over 2} \\
2 & = -5 + r \\
2 + 5 & = r \\
7 & = r \\
\\
\therefore & \phantom{.} R(7, 0)
\end{align}
\begin{align}
x & = -5
&& \text{ or } &
x & = 7 \\
x + 5 & = 0
&&&
x - 7 & = 0
\end{align}
\begin{align}
0 & = (x + 5)(x - 7) \\
0 & = x^2 - 7x + 5x - 35 \\
0 & = x^2 - 2x - 35 \\
\\
\text{Eqn of curve: } y & = -(x^2 - 2x - 35) \\
y & = - x^2 + 2x + 35 \\
\\
\text{Let } & x - 0, \\
y & = -(0)^2 + 2(0) + 35 \\
y & = 35 \\
\\
\therefore & \phantom{.} Q(0, 35)
\end{align}
(b)
\begin{align} \text{Line of symmetry, } & x = 1 \\ \\ \text{Substitute } x = 1 & \text{ into eqn of curve,} \\ y & = -(1)^2 + 2(1) + 35 \\ y & = 36 \\ \\ \text{Max. point: } & (1, 36) \end{align}
(c)
\begin{align} \text{Substitute } x = -3 & \text{ into eqn of curve,} \\ y & = -(-3)^2 + 2(-3) + 35 \\ y & = 20 \\ \\ \therefore k & = 20 \end{align}
(a)
\begin{align} x + 7 \end{align}
(b)
\begin{align} \text{Product} & = x \times (x + 7) \\ y & = x(x + 7) \\ y & = x^2 + 7x \end{align}
(c)
\begin{align} y & = x^2 + 7x \phantom{0000000000} [\text{Minimum curve } \cup] \\ y & = x^2 + 7x + \left(7 \over 2\right)^2 - \left(7 \over 2\right)^2 \\ y & = x^2 + 7x + 3.5^2 - 3.5^2 \\ y & = (x + 3.5)^2 - 12.25 \\ \\ \text{Turning } & \text{point: } (-3.5, -12.25) \\ \\ \text{Let } & x = 0, \\ y & = (0 + 3.5)^2 - 12.25 \\ y & = 0 \phantom{000000000000000000000} [y \text{-intercept}] \end{align}
(d)
\begin{align} \text{Turing point: } & (-3.5, -12.25) \\ \\ \text{Min. value of product } (y) & = -12.25 \\ \\ \text{Smaller number } (x) & = -3.5 \\ \\ \text{Larger number} & = x + 7 \\ & = -3.5 + 7 \\ & = 3.5 \end{align}
(a)
(x = 20 is the line of symmetry)
\begin{align} \text{Distance from } O & = 2 \times 20 \\ & = 40 \text{ m} \end{align}
(b)
\begin{align} x & = 0 &&& x & = 40 \\ & &&& x - 40 & = 0 \end{align} \begin{align} 0 & = x(x - 40) \\ \\ \text{Eqn of curve: } & y = a x (x - 40) \\ \\ \text{Using} & \text{ max. point } (20, 5), \\ 5 & = a(20)(20 - 40) \\ 5 & = a(20)(-20) \\ 5 & = -400a \\ {5 \over -400} & = a \\ -{1 \over 80} & = a \\ \\ \text{Eqn of curve: } & y = -{1 \over 80} x (x - 40) \\ \end{align}
(a)
\begin{align} y & = 2 + 4x - x^2 \\ \\ \text{Let } & x = 0, \\ y & = 2 + 4(0) - (0)^2 \\ y & = 2 \\ \\ OA & = 2 \text{ m} \end{align}
(b)
\begin{align}
y & = 2 + 4x - x^2 \\
\\
\text{Let } & y = 2, \\
2 & = 2 + 4x - x^2 \\
2 - 2 & = 4x - x^2 \\
0 & = 4x - x^2 \\
0 & = x(4 - x)
\end{align}
\begin{align}
x & = 0
&& \text{ or } &
4 - x & = 0 \\
& &&&
-x & = -4 \\
& &&&
x & = 4
\end{align}
\begin{align}
\therefore C(4, 2) & \text{ and } D(4, 0) \\
\\
OD & = 4 \text{ m}
\end{align}
(c)
\begin{align} \text{Line of symmetry, } x & = {0 + 4 \over 2} \\ x & = 2 \\ \\ \text{Substitute } x = 2 & \text{ into } y = 2 + 4x - x^2, \\ y & = 2 + 4(2) - (2)^2 \\ y & = 6 \\ \\ \therefore & \phantom{.} B(2, 6) \end{align}
(d)
\begin{align} y & = 2 + 4x - x^2 \\ y & = - x^2 + 4x + 2 \phantom{000000} [\text{Maximum curve } \cap ] \\ \\ y & \text{-intercept: } 2 \\ \\ x & \text{-intercepts: } 0, 4 \\ \\ \text{Tur} & \text{ning point: } (2, 6) \end{align}
(a)
\begin{align} y & = \pm (x - h)^2 + k \\ \\ \text{Since tur} & \text{ning point is } (3, - 1), \\ y & = \pm (x - 3)^2 - 1 \end{align}
(b)
\begin{align} y & = (x - 3)^2 - 1 \phantom{000000} [\text{Minimum curve } \cup] \\ \\ \text{Turning} & \text{ point: } (3, -1) \\ \\ \text{Let } & x = 0, \\ y & = (0 - 3)^2 - 1 \\ y & = 8 \phantom{000000000000000} [y \text{-intercept}] \end{align}
\begin{align} y & = -(x - 3)^2 - 1 \phantom{000000} [\text{Maximum curve } \cap] \\ \\ \text{Turning} & \text{ point: } (3, -1) \\ \\ \text{Let } & x = 0, \\ y & = -(0 - 3)^2 - 1 \\ y & = -10 \phantom{000000000000000} [y \text{-intercept}] \end{align}
(a)
\begin{align} y & = (x - 2)^2 + (c - 4) \\ \\ \text{Turning} & \text{ point: } (2, c - 4) \end{align} \begin{align} h & = 2 &&& c - 4 & = -5 \\ & &&& c & = -5 + 4 \\ & &&& c & = -1 \end{align}
(b)
\begin{align} y & = (x - 2)^2 + (c - 4) \\ y & = (x - 2)^2 + (-1 - 4) \\ y & = (x - 2)^2 - 5 \\ \\ \text{Minimum} & \text{ point: } (2, -5) \\ \\ \text{Let } & x = 0, \\ y & = (0 - 2)^2 - 5 \\ y & = -1 \phantom{000000000000} [y \text{-intercept}] \end{align}
(a)
\begin{align} & \text{Not possible for } y \text{-intercept to be positive if } x \text{-intercepts are -2 and 3} \\ \\ & \therefore \text{Alice and Casey are correct} \end{align}
(b)
\begin{align} y & = x^2 - bx + c \\ \\ \text{Using } & (0, 4), \\ 4 & = (0)^2 - b(0) + c \\ 4 & = 0 - 0 + c \\ 4 & = c \\ \\ y & = x^2 - bx + 4 \\ y & = x^2 - bx + \left(b \over 2\right)^2 - \left(b \over 2\right)^2 + 4 \\ y & = \left(x - {b \over 2}\right)^2 - {b^2 \over 4} + 4 \\ \\ \text{Minimum} & \text{ point: } \left({b \over 2}, -{b^2 \over 4} + 4\right) \\ \\ \therefore {b \over 2} & = 1 \\ {b \over 2} & = {2 \over 2} \\ b & = 2 \\ \\ \\ \therefore b & = 2, c = 4 \end{align}