MATHEMATICS
Grade 11
MORE EXERCISES
Quadratic graphs, parabolae : answers.
  
  
Answers  1
    
$$ \hspace*{2 mm}\mathrm{1.1\kern3mmf(x) = x^2 + 4x + 3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = x^2 + 4x + 3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ \ \\ y = (0)^2 + 4(0) + 3\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{= 3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ is\ (0 ; 3)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ \ :\ x^2 + 4x + 3 = 0\kern2mm\ } $$ $$ \hspace*{37 mm}\mathrm{(x + 1)(x + 3) = 0\kern2mm\ } $$ $$ \hspace*{37 mm}\mathrm{x = −1\ \ or\ \ x = −3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ are\ (−3 ; 0)\ and\ \ (−1 ; 0)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Axis\ of\ simmetry\ x = \frac{−b}{2a}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= \frac{−4}{2 \times 1}\kern2mm\ } $$
$$ \hspace*{43 mm}\mathrm{= −2\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = (−2)^2 + 4(−2) + 3\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −1\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (−2 ; −1)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 + 4x + 3 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 + 4x + (\frac{4}{2})^2 − (\frac{4}{2})^2 + 3 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{(x + 2)^2 − 1 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (−2 ; −1)\kern2mm\ } $$
  

                                                                   [ Q 1.1 ]
    
$$ \hspace*{2 mm}\mathrm{1.2\kern3mmf(x) = x^2 − 2x − 3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = x^2 − 2x − 3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ \ :\ y = (0)^2 − 2(0) − 3\kern2mm\ } $$ $$ \hspace*{41mm}\mathrm{= −3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ is\ (0 ; −3)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ \ :\ x^2 − 2x − 3 = 0\kern2mm\ } $$ $$ \hspace*{37 mm}\mathrm{(x + 1)(x − 3) = 0\kern2mm\ } $$ $$ \hspace*{37 mm}\mathrm{x = −1\ \ of\ \ x = 3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ are\ (−1 ; 0)\ and\ \ (3 ; 0)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Axis\ of\ simmetry\ x = \frac{−b}{2a}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= \frac{−(−2)}{2 \times 1}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= 1\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = (1)^2 − 2(1) − 3\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −4\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (1 ; −4)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 − 2x − 3 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 − 2x + (\frac{−2}{2})^2 − (\frac{−2}{2})^2 − 3 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{(x − 1)^2 − 4 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (1 ; −4)\kern2mm\ } $$
  

                                                                   [ Q 1.2 ]
    
$$ \hspace*{2 mm}\mathrm{1.3\kern3mmf(x) = x^2 − 3x + 2\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = x^2 − 3x + 2\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ \ :\ y = (0)^2 − 3(0) + 2\kern2mm\ } $$ $$ \hspace*{41 mm}\mathrm{= 2\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ is\ (0 ; 2)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ \ :\ x^2 − 3x +2 = 0\kern2mm\ } $$ $$ \hspace*{37 mm}\mathrm{(x − 1)(x − 2) = 0\kern2mm\ } $$ $$ \hspace*{37 mm}\mathrm{x = 1\ \ of\ \ x = 2\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ are\ (1 ; 0)\ and\ \ (2 ; 0)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Axis\ of\ simmetry\ x = \frac{−b}{2a}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= \frac{−(−3)}{2 \times 1}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= 1,5\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = (1,5)^2 − 3(1,5) + 2\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −0,25\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (1,5 ; −0,25)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 − 3x + 2 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 − 3x + (\frac{−3}{2})^2 − (\frac{−3}{2})^2 + 2= 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{(x − 1,5)^2 − 0,25 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (1 ; −0,25)\kern2mm\ } $$
  

                                                                   [ Q 1.3 ]
    
$$ \hspace*{2 mm}\mathrm{1.4\kern3mmf(x) = x^2 + 2x − 3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = x^2 + 2x − 3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ \ :\ y = (0)^2 + 2(0) − 3\kern2mm\ } $$ $$ \hspace*{41 mm}\mathrm{= −3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ is\ D(0 ; −3)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ \ :\ x^2 + 2x − 3 = 0\kern2mm\ } $$ $$ \hspace*{37 mm}\mathrm{(x − 1)(x + 3) = 0\kern2mm\ } $$ $$ \hspace*{37 mm}\mathrm{x = 1\ \ or\ \ x = −3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ are\ B(1 ; 0)\ and\ \ A(−3 ; 0)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Axis\ of\ simmetry\ x = \frac{−b}{2a}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= \frac{−(2)}{2 \times 1}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= −1\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = (−1)^2 + 2(−1) − 3\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −4\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (−1 ; −4)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 + 2x − 3= 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 + 2x + (\frac{2}{2})^2 − (\frac{2}{2})^2 − 3 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{(x + 1)^2 − 4 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (−1 ; −4)\kern2mm\ } $$
  

                                                                   [ Q 1.4 ]
    
$$ \hspace*{2 mm}\mathrm{1.5\kern3mmf(x) = −x^2 − 5x − 4\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = −x^2 − 5x − 4\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ \ :\ y = −(0)^2 − 5(0) − 4\kern2mm\ } $$ $$ \hspace*{41 mm}\mathrm{= −4\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ is\ D(0 ; −4)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ \ :\ −x^2 − 5x − 4 = 0\kern2mm\ } $$ $$ \hspace*{42 mm}\mathrm{x^2 + 5x + 4 = 0\kern2mm\ } $$ $$ \hspace*{39 mm}\mathrm{(x + 1)(x + 4) = 0\kern2mm\ } $$ $$ \hspace*{39 mm}\mathrm{x = −1\ \ or\ \ x = −4\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ are\ B(−1 ; 0)\ and\ \ A(−4 ; 0)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Axis\ of\ simmetry\ x = \frac{−b}{2a}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= \frac{−(−5)}{2 \times −1}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= −2,5\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = −(−2,5)^2 − 5(−2,5) − 4\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= 2,5\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (−2,5 ; 2,5)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{−x^2 − 5x − 4 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 + 5x + 4 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 + 5x + (\frac{5}{2})^2 − (\frac{5}{2})^2 + 4 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{(x + 2,5)^2 + 2,5 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (−2,5 ; 2,5)\kern2mm\ } $$
  

                                                                   [ Q 1.5 ]
    
$$ \hspace*{2 mm}\mathrm{1.6\kern3mmf(x) = −x^2 − 3x + 10\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = −x^2 − 3x + 10\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Y-intercept\ is\ D(0 ; 10)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ \ :\ −x^2 − 3x + 10 = 0\kern2mm\ } $$ $$ \hspace*{42 mm}\mathrm{x^2 + 3x − 10 = 0\kern2mm\ } $$ $$ \hspace*{41 mm}\mathrm{(x + 5)(x − 2) = 0\kern2mm\ } $$ $$ \hspace*{45 mm}\mathrm{x = −5\ \ or\ \ x = 2\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{X-intercepts\ are\ A(−5 ; 0)\ and\ \ B(2 ; 0)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Axis\ of\ simmetry\ x = \frac{−b}{2a}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= \frac{−(−3)}{2 \times −1}\kern2mm\ } $$

$$ \hspace*{43 mm}\mathrm{= −1,5\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y = −(−1,5)^2 − 3(−1,5) + 10\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= 12,25\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (−1,5 ; 12,25)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{−x^2 − 3x + 10 = 0\kern2mm\ } $$
$$ \hspace*{14 mm}\mathrm{x^2 + 3x − 10 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x^2 + 3x + (\frac{3}{2})^2 − (\frac{3}{2})^2 − 10 = 0\kern2mm\ } $$
$$ \hspace*{24 mm}\mathrm{(x + 1,5)^2 + 12,25 = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{Turning\ point\ is\ (−1,5 ; 12,25)\kern2mm\ } $$
  

                                                                   [ Q 1.6 ]
  
Answers  2
    
$$ \hspace*{2 mm}\mathrm{2.1\kern3mmy = x^2 + 5x − 2\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= x^2 + 5x + \Big(\frac{5}{2}\Big)^2 − \Big(\frac{5}{2}\Big)^2 − 2\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= \Big(x + \frac{5}{2}\Big)^2 − \frac{33}{4}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Turning\ point\ is\ \Big(−\frac{5}{2} ; −\frac{33}{2}\Big)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Y-intercept\ is\ (0 ; −2)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{X-intercepts\ \ :\ \Big(x + \frac{5}{2}\Big)^2 − \frac{33}{4} = 0\kern2mm\ } $$ $$ \hspace*{33 mm}\mathrm{x = \Big(− \frac{5}{2}\Big) \plusmn \Big(\sqrt{\frac{33}{4}}\Big)\kern2mm\ } $$ $$ \hspace*{33 mm}\mathrm{x = − 5,37\ or\ \ x = 0,37\kern2mm\ } $$ $$ \hspace*{4 mm}\mathrm{X-intercepts\ are\ (−5,37 ; 0)\ \ and\ \ (0,37 ; 0)\kern2mm\ } $$

  
                                                                   [ Q 2.1 ]
    
$$ \hspace*{2 mm}\mathrm{2.2\kern3mmy = x^2 − 2x − 4\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= x^2 − 2x + \Big(\frac{− 2}{2}\Big)^2 − \Big(\frac{− 2}{2}\Big)^2 − 4\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= (x − 1)^2 − 5\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Turning\ point\ is\ (1 ; −5)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Y-intercept\ is\ (0 ; −4)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{X-intercepts\ \ :\ (x − 1)^2 − 5 = 0\kern2mm\ } $$ $$ \hspace*{52 mm}\mathrm{x = 1 \plusmn \sqrt{5}\kern2mm\ } $$ $$ \hspace*{33 mm}\mathrm{x = − 1,24\ or\ \ x = 3,24\kern2mm\ } $$ $$ \hspace*{12 mm}\mathrm{X-intercepts\ are\ A(−1,24 ; 0)\ and\kern2mm\ } $$ $$ \hspace*{40 mm}\mathrm{B(3,24 ; 0)\kern2mm\ } $$

  
                                                                   [ Q 2.2 ]
    
$$ \hspace*{2 mm}\mathrm{2.3\kern3mmy = 2x^2 − x − 3\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= 2(x^2 − \frac{1}{2}x − \frac{3}{2})\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= 2\Big(x^2 − \frac{1}{2}x + \Big(\frac{−1}{2 \times 2}\Big)^2 − \Big(\frac{−1}{2 \times 2}\Big)^2 − \frac{3}{2}\Big)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= 2\Big(\Big(x − \frac{1}{4}\Big)^2 − \frac{25}{16}\Big)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= 2\Big(x − \frac{1}{4}\Big)^2 − \frac{25}{8}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Turning\ point\ is\ \Big(\frac{1}{4} ; −\frac{25}{8}\Big)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Y-intercept\ is\ (0 ; −3)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{X-intercepts\ \ :\ 2\Big(x − \frac{1}{4}\Big)^2 − \frac{25}{8} = 0\kern2mm\ } $$ $$ \hspace*{40 mm}\mathrm{x = \frac{1}{4} \plusmn \Big(\sqrt{\frac{25}{16}}\Big)\kern2mm\ } $$ $$ \hspace*{40 mm}\mathrm{x = − 1\ of\ \ x = 1,5\kern2mm\ } $$ $$ \hspace*{10 mm}\mathrm{X-intercepts\ are\ A(−1 ; 0)\ \ or\ \ B(1,5 ; 0)\kern2mm\ } $$

  
                                                                   [ Q 2.3 ]
    
$$ \hspace*{2 mm}\mathrm{2.4\kern3mmy = 3x^2 − 2x − 5\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= 3(x^2 − \frac{2}{3}x − \frac{5}{3})\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= 3\Big(x^2 − \frac{2}{3}x + \Big(\frac{−2}{3 \times 2}\Big)^2 − \Big(\frac{−2}{3 \times 2}\Big)^2 − \frac{5}{3}\Big)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= 3\Big(\Big(x − \frac{1}{3}\Big)^2 − \frac{16}{9}\Big)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= 3\Big(x − \frac{1}{3}\Big)^2 − \frac{16}{3}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Turning\ point\ C\ is\ \Big(\frac{1}{3} ; −\frac{16}{3}\Big)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Y-intercept\ is\ (0 ; −5)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{X-intercepts\ \ :\ 3\Big(x − \frac{1}{3}\Big)^2 − \frac{16}{3} = 0\kern2mm\ } $$ $$ \hspace*{39 mm}\mathrm{x = \frac{1}{3} \plusmn \Big(\sqrt{\frac{16}{9}}\Big)\kern2mm\ } $$ $$ \hspace*{39 mm}\mathrm{x = − 1\ or\ \ x = \frac{5}{3}\kern2mm\ } $$ $$ \hspace*{10 mm}\mathrm{X-intercepts\ are\ A(−1 ; 0)\ \ or\ \ B(\frac{5}{3} ; 0)\kern2mm\ } $$

  
                                                                   [ Q 2.4 ]
    
$$ \hspace*{2 mm}\mathrm{2.5\kern3mmy = −x^2 − x + 4\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{y = −(x^2 + x − 4)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= −(x^2 + x + \Big(\frac{1}{2}\Big)^2 − \Big(\frac{1}{2}\Big)^2 − 4 )\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= −\Big(\Big(x + \frac{1}{2}\Big)^2 − \frac{17}{4}\Big)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= −\Big(x + \frac{1}{2}\Big)^2 + \frac{17}{4}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Turning\ point\ C\ is\ \Big(−\frac{1}{2} ; \frac{17}{4}\Big)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Y-intercept\ is\ D(0 ; 4)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{X-intercepts\ \ :\ −\Big(x + \frac{1}{2}\Big)^2 + \frac{17}{4} = 0\kern2mm\ } $$ $$ \hspace*{39 mm}\mathrm{x = \frac{1}{2} \plusmn \Big(\sqrt{\frac{17}{4}}\Big)\kern2mm\ } $$ $$ \hspace*{39 mm}\mathrm{x = − 2,56\ or\ \ x = 1,56\kern2mm\ } $$ $$ \hspace*{10 mm}\mathrm{X-intercepts\ are\ A(−2,56 ; 0)\ \ or\ \ B(1,56 ; 0)\kern2mm\ } $$

  
                                                                   [ Q 2.5 ]
    
$$ \hspace*{2 mm}\mathrm{2.6\kern3mmy = −2x^2 + 3x − 5\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{y = −2\Big(x^2 − \frac{3}{2}x + \frac{5}{2}\Big)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= −2\Big(x^2 − \frac{3}{2}x + \Big(\frac{−3}{2 \times 2}\Big)^2 − \Big(\frac{−3}{2 \times 2}\Big)^2 + \frac{5}{2} \Big)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= −2\Big(\Big(x − \frac{3}{4}\Big)^2 + \frac{31}{16}\Big)\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{= −2\Big(x − \frac{3}{4}\Big)^2 − \frac{31}{8}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Turning\ point\ C\ is\ \Big(\frac{3}{4} ; −\frac{31}{8}\Big)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Y-intercept\ is\ D(0 ; −5)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{X-intercepts\ \ :\ −2\Big(x − \frac{3}{4}\Big)^2 − \frac{31}{8} = 0\kern2mm\ } $$ $$ \hspace*{39 mm}\mathrm{\Big(x − \frac{3}{4}\Big)^2 = − \frac{31}{16}\kern2mm\ } $$ $$ \hspace*{47 mm}\mathrm{x = \frac{3}{4} \plusmn \Big(\sqrt{\frac{−31}{16}}\Big)\kern2mm\ } $$ $$ \hspace*{8 mm}\mathrm{No\ real\ roots\ and\ thus\ no\ X-intercepts.\kern2mm\ } $$
  
                                                                   [ Q 2.6 ]
  
Answers  3
    
   3.1  Let y = ax2 +bx + c
          Make an equation by substituting the
          coordinates of a point. Solve the formed
          simultaneous equations to determine
          values of a, b and c, and thus
          the equation.
          Y-intercept is (0 ; −2) and thus c = −2
          The equation is now y = ax2 + bx − 2
$$ \hspace*{10 mm}\mathrm{At\ A(−2;0)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(−2)^2 + b(−2) − 2 = 0\kern2mm\ } $$
$$ \hspace*{42 mm}\mathrm{4a −2b = 2\ \kern2mm\ (1)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ B(1;0)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(1)^2 + b(1) − 2 = 0\kern2mm\ } $$
$$ \hspace*{41 mm}\mathrm{a + b = 2\ \kern2mm\ (2)\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{Solve\ the\ simultaneous\ equations\ :\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(2) \times 2\ :\ 2a + 2b = 2\ \kern2mm\ (3)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(1) + (3)\ :\ \kern4mm\ 6a = 6\kern2mm\ } $$
$$ \hspace*{42 mm}\mathrm{\therefore\ a = 1\kern2mm\ } $$
$$ \hspace*{21 mm}\mathrm{Into\ (2)\ :\ 1 + b = 2\kern2mm\ } $$
$$ \hspace*{42 mm}\mathrm{\therefore\ b = 1\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{\therefore\ y = x^2 + x − 2\kern2mm\ } $$             [ Q 3.1 ]
$$ \hspace*{42 mm}\mathrm{\bold{OR}\kern2mm\ } $$
          Substitute the coordinates of the
          X-intercepts and thus the roots of the
          equation into the equation
          y = a(x + one root)(x + second root).
          Simplify. Substitute the coordintes of
          a third point and solve.

          X-intercepts are (−2;0) and (1;0)
$$ \hspace*{33 mm}\mathrm{y = a(x + 2)(x − 1)\kern2mm\ } $$
$$ \hspace*{36 mm}\mathrm{= ax^2 + ax − 2a \kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (0 ; −2)\ :\ \ −2 = a(0)^2 + a(0) − 2a \kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{−2 = − 2a \kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{\therefore\ a = 1\kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{\therefore\ y = x^2 + x − 2\kern2mm\ } $$     [ Q 3.1 ]
    
   3.2  Let y = ax2 +bx + c
          Y-intercept is (0 ; 6) and thus c = 6
          The equation is now y = ax2 + bx + 6
$$ \hspace*{10 mm}\mathrm{At\ A(2;0)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(2)^2 + b(2) + 6 = 0\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{4a + 2b = − 6\ \kern2mm\ (1)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ B(3;0)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(3)^2 + b(3) + 6 = 0\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{9a + 3b = −6\ \kern2mm\ (2)\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{Solve\ the\ simultaneoaus\ equations\ :\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(1) \times 3\ :\ 12a + 6b = −18\ \kern2mm\ (3)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(2) \times 2\ :\ 18a + 6b = −12\ \kern2mm\ (4)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(4) − (3)\ :\ \kern4mm\ 6a = 6\kern2mm\ } $$
$$ \hspace*{42 mm}\mathrm{\therefore\ a = 1\kern2mm\ } $$
$$ \hspace*{21 mm}\mathrm{Into\ (1)\ :\ 4(1) + 2b = −6\kern2mm\ } $$
$$ \hspace*{45 mm}\mathrm{\therefore\ b = −5\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{\therefore\ y = x^2 − 5x + 6\kern2mm\ } $$      [ Q 3.2 ]
$$ \hspace*{42 mm}\mathrm{\bold{OR}\kern2mm\ } $$
          X-intercepts are (2;0) and (3;0)
$$ \hspace*{33 mm}\mathrm{y = a(x − 2)(x − 3)\kern2mm\ } $$
$$ \hspace*{36 mm}\mathrm{= ax^2 − 5ax + 6a \kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (0 ; 6)\ :\ \ 6 = a(0)^2 − 5a(0) + 6a \kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{6 = 6a \kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{\therefore\ a = 1\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{\therefore\ y = x^2 − 5x + 6\kern2mm\ } $$      [ Q 3.2 ]
  
Answers  4
    
   4.1  Let y = ax2 +bx + c
          Y-intercept is (0 ; 3) and thus c = 3
          Equation is now y = ax2 + bx + 3
$$ \hspace*{10 mm}\mathrm{At\ (−1;0)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(−1)^2 + b(−1) + 3 = 0\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{a − b = − 3\ \kern2mm\ (1)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (2;15)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(2)^2 + b(2) + 3 = 15\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{4a + 2b = 12\ \kern2mm\ (2)\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{Solve\ simultaneous\ equations\ :\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(1) \times 2\ :\ 2a − 2b = −6\ \kern2mm\ (3)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(2) + (3)\ :\ \kern4mm\ 6a = 6\kern2mm\ } $$
$$ \hspace*{42 mm}\mathrm{\therefore\ a = 1\kern2mm\ } $$
$$ \hspace*{21 mm}\mathrm{Into\ (1)\ :\ (1) − b = −3\kern2mm\ } $$
$$ \hspace*{41 mm}\mathrm{\therefore\ b = 4\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{\therefore\ y = x^2 + 4x + 3\kern2mm\ } $$          [ Q 4.1 ]

    
   4.2  Let y = ax2 +bx + c
$$ \hspace*{10 mm}\mathrm{At\ (−3;−12)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(−3)^2 + b(−3) + c = −12\kern2mm\ } $$
$$ \hspace*{35 mm}\mathrm{9a − 3b + c = − 12\ \kern2mm\ (1)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (2;−7)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(2)^2 + b(2) + c = −7\kern2mm\ } $$
$$ \hspace*{32 mm}\mathrm{4a + 2b + c = −7\ \kern2mm\ (2)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (5;20)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(5)^2 + b(5) + c = 20\kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{25a + 5b + c = 20\ \kern2mm\ (3)\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{Solve\ simultaneous\ equations\ :\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{Eliminate\ c\ first\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(2) − (1)\ :\ 5a − 5b = −5\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{Simplify\ :\ \ \ \ \ a − b = −1\ \kern2mm\ (4)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(3) − (1)\ :\ 16a + 8b = 32\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{Simplify\ :\ \ \ \ \ 2a + b = 4\ \kern2mm\ (5)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(4) + (5)\ :\ 16a + 8b = 32\kern2mm\ } $$
$$ \hspace*{50 mm}\mathrm{3a = 3\kern2mm\ } $$
$$ \hspace*{48 mm}\mathrm{\therefore\ a = 1\kern2mm\ } $$
$$ \hspace*{21 mm}\mathrm{Into\ (5)\ :\ \ \ \ \ \ 2(1) + b = 4\kern2mm\ } $$
$$ \hspace*{47 mm}\mathrm{\therefore\ b = 2\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{Into\ (1)\ :\ 9(1) − 3(2) + c = −12\kern2mm\ } $$
$$ \hspace*{48 mm}\mathrm{\therefore\ c = −15\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{\therefore\ y = x^2 + 2x − 15\kern2mm\ } $$        [ Q 4.2 ]
    
   4.3  Let y = ax2 +bx + c
$$ \hspace*{10 mm}\mathrm{At\ (−2;−15\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(−2)^2 + b(−2) + c = −15\kern2mm\ } $$
$$ \hspace*{35 mm}\mathrm{4a − 2b + c = − 15\ \kern2mm\ (1)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (1;0)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(1)^2 + b(1) + c = 0\kern2mm\ } $$
$$ \hspace*{32 mm}\mathrm{a + b + c = 0\ \kern2mm\ (2)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (4;−3)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(4)^2 + b(4) + c = −3\kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{16a + 4b + c = −3\ \kern2mm\ (3)\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{Solve\ simultaneous\ equations\ :\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{Eliminate\ c\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(1) − (2)\ :\ 3a − 3b = −15\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{Simplify\ :\ \kern3mm\ a − b = −5\ \kern2mm\ (4)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(3) − (1)\ :\ 12a + 6b = 12\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{Simplify\ :\ \kern3mm\ 2a + b = 2\ \kern2mm\ (5)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(4) + (5)\ :\ \kern9mm\ 3a = −3\kern2mm\ } $$
$$ \hspace*{48 mm}\mathrm{\therefore\ a = −1\kern2mm\ } $$
$$ \hspace*{21 mm}\mathrm{Into\ (4)\ :\ \ \ (−1) − b = −5\kern2mm\ } $$
$$ \hspace*{45 mm}\mathrm{\therefore\ b = 4\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{Into\ (2)\ :\ (−1) + 4 + c = 0\kern2mm\ } $$
$$ \hspace*{45 mm}\mathrm{\therefore\ c = −3\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{\therefore\ y = −x^2 + 4x − 3\kern2mm\ } $$      [ Q 4.3 ]
    
   4.4  Let y = ax2 +bx + c
$$ \hspace*{10 mm}\mathrm{At\ (−3;−14)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(−3)^2 + b(−3) + c = −14\kern2mm\ } $$
$$ \hspace*{35 mm}\mathrm{9a − 3b + c = − 14\ \kern2mm\ (1)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (−1;4)\ :\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{a(−1)^2 + b(−1) + c = 4\kern2mm\ } $$
$$ \hspace*{39 mm}\mathrm{a − b + c = 4\ \kern2mm\ (2)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (2;1)\ :\kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{a(2)^2 + b(2) + c = 1\kern2mm\ } $$
$$ \hspace*{35 mm}\mathrm{4a + 2b + c = 1\ \kern2mm\ (3)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(1) − (2)\ :\ 8a − 2b = −18\kern2mm\ } $$
$$ \hspace*{42 mm}\mathrm{4a − b = −9\ \kern2mm\ (4)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{(1) − (3)\ :\ 5a − 5b = −15\kern2mm\ } $$
$$ \hspace*{44 mm}\mathrm{a − b = −3\ \kern2mm\ (5)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{(4) + (5)\ :\ 3a = −3\kern2mm\ } $$
$$ \hspace*{45 mm}\mathrm{\therefore\ a = −1\kern2mm\ } $$
$$ \hspace*{21 mm}\mathrm{Into\ (4)\ :\ \ \ (−1) − b = −5\kern2mm\ } $$
$$ \hspace*{45 mm}\mathrm{\therefore\ b = 4\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{Into\ (2)\ :\ (−1) + 4 + c = 0\kern2mm\ } $$
$$ \hspace*{45 mm}\mathrm{\therefore\ c = −3\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{\therefore\ y = −x^2 + 4x − 3\kern2mm\ } $$       [ Q 4.4 ]
    
   4.5  Let y = a(x + p)2 + q
          Turning point is (2 ; −1)
$$ \hspace*{25 mm}\mathrm{y = a(x − 2)^2 −1\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (4;1)\ : a(4 − 2)^2 −1 = 1\kern2mm\ } $$
$$ \hspace*{42 mm}\mathrm{4a = 2\kern2mm\ } $$
$$ \hspace*{44 mm}\mathrm{a = 0,5\kern2mm\ } $$
$$ \hspace*{40 mm}\mathrm{\therefore\ y = 0,5(x − 2)^2 − 1\kern2mm\ } $$
$$ \hspace*{40 mm}\mathrm{\therefore\ y = 0,5x^2 − 2x + 1\kern2mm\ } $$
                                                                   [ Q 4.5 ]

    
   4.6  Let y = a(x + p)2 + q
          Turning point is (1 ; 4)
$$ \hspace*{25 mm}\mathrm{y = a(x − 1)^2 + 4\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (−3;−12)\ : a(−3 − 1)^2 + 4 = −12\kern2mm\ } $$
$$ \hspace*{48 mm}\mathrm{16a = −16\kern2mm\ } $$
$$ \hspace*{52 mm}\mathrm{a = −1\kern2mm\ } $$
$$ \hspace*{40 mm}\mathrm{\therefore\ y = −(x − 1)^2 + 4\kern2mm\ } $$
$$ \hspace*{40 mm}\mathrm{\therefore\ y = −x^2 + 2x + 3\kern2mm\ } $$
                                                                   [ Q 4.6 ]

    
   4.7  Let y = a(x + p)2 + q
$$ \hspace*{10 mm}\mathrm{Turning\ point\ is\ \Big(−\frac{1}{3} ; \frac{28}{3}\Big)\kern2mm\ } $$

$$ \hspace*{25 mm}\mathrm{y = a\Big(x + \frac{1}{3}\Big)^2 + \frac{28}{3}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ (−4;−31)\ : a\Big(−4 + \frac{1}{3}\Big)^2 + \frac{28}{3} = −31\kern2mm\ } $$

$$ \hspace*{44 mm}\mathrm{\frac{121}{9}a + \frac{28}{3} = −31\kern2mm\ } $$
$$ \hspace*{46 mm}\mathrm{121a + 84 = −279\kern2mm\ } $$
$$ \hspace*{46 mm}\mathrm{a = −3\kern2mm\ } $$
$$ \hspace*{40 mm}\mathrm{\therefore\ y = −3(x + \frac{1}{3})^2 + \frac{28}{3}\kern2mm\ } $$
$$ \hspace*{40 mm}\mathrm{\therefore\ y = −3x^2 − 6x + 9\kern2mm\ } $$
                                                                      [ Q 4.7 ]

  
Answers  5
    
   5.1  y = x2 + 2x − 8
          C is the Y-intercept. c = −8 and thus
          C (0 ; −8) A and B are the X-intercepts
          and thus is y = 0
$$ \hspace*{13 mm}\mathrm{x^2 + 2x − 8 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{(x + 4)(x − 2) = 0\kern2mm\ } $$
                x = 4 or x = 2
         A is the point (−4 ; 0) and
         B is the point (2 ; 0)
         D is the turning point and thus it is on
         the axis of simmetry.
$$ \hspace*{10 mm}\mathrm{At\ D\ x = \frac{−b}{2a}\kern2mm\ } $$
$$ \hspace*{18 mm}\mathrm{x = \frac{−2}{2(1)} = −1\kern2mm\ } $$

$$ \hspace*{18 mm}\mathrm{y = (−1)^2 + 2(−1) − 8\kern2mm\ } $$
$$ \hspace*{21 mm}\mathrm{= −9\kern2mm\ } $$
         D is the point (−1 ; −9)                    [ Q 5.1 ]
    
   5.2  M is a point on the axis of simmetry and
          is the point (−1 ; 0)
$$ \hspace*{10 mm}\mathrm{AM = x_M − x_A\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{= (−1) − (− 4)\ =\ 3\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{DM = y_M − y_D\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{= (0) − (− 9)\ =\ 9\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{OM = x_O − x_M\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{= (0) − (− 1)\ =\ 1\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{OC = y_O − y_C\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{= (0) − (− 8)\ =\ 8\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{AC = \sqrt{AO^2 + OC^2}\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{= \sqrt{(4)^2 + (8)^2}\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{= \sqrt{80}\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{= 4\sqrt{5} = 8,944\kern2mm\ } $$                [ Q 5.2 ]
    
$$ \hspace*{2 mm}\mathrm{5.3.1\kern2mmAt\ P(−7;p)\ : y = x^2 + 2x − 8\kern2mm\ } $$
$$ \hspace*{32 mm}\mathrm{p = (−7)^2 + 2 \times (−7) − 8\kern2mm\ } $$
$$ \hspace*{34 mm}\mathrm{= 27\kern2mm\ } $$             [ Q 5.3.1 ]
    
   5.3.2  N is the point (−1 ; 27) and
             Q is the point (5 ; 27)                   [ Q 5.3.2 ]
    
$$ \hspace*{2 mm}\mathrm{5.3.3\kern2mmNQ\ has\ a\ length\ of\ d(NQ) = x_Q − x_N\kern2mm\ } $$
$$ \hspace*{42 mm}\mathrm{d(NQ) = 6 − (−1)\kern2mm\ } $$
$$ \hspace*{52 mm}\mathrm{= 7\kern2mm\ } $$
$$ \hspace*{42 mm}\mathrm{d(ND) = y_N − y_D\kern2mm\ } $$
$$ \hspace*{52 mm}\mathrm{= 27 − (−9)\kern2mm\ } $$
$$ \hspace*{52 mm}\mathrm{= 36\kern2mm\ } $$
                                                                   [ Q 5.3.3 ]
    
   5.4  W is a point on RS and ND. Thus W is
          on the axis of simmetry and RW = WS
          so that RW + WS = 4. Therefore R is
          4 units left of W and S is
          4 units to the right of W. Therefore R is the
          point (-5 ; yR) and S the point (3 ; yS).
          Now determine the values of the
          y-coordinates of R and S.
$$ \hspace*{8 mm}\mathrm{At\ R(−5;r)\ : y = x^2 + 2x − 8\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{r = (−5)^2 + 2 \times (−5) − 8\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= 7\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{At\ S(3;s)\ : y = x^2 + 2x − 8\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{r = 3^2 + 2 \times 3 − 8\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= 7\kern2mm\ } $$
          R is the point (-5 ; 7) and S is
          the point (3 ; 7)                             [ Q 5.4 ]
    
   5.5  P is the point (-7 ; ) and W the
          point (−1 ; 7). Length of PW = d(PW)
$$ \hspace*{9 mm}\mathrm{d(PW) = \sqrt{PN^2 + NW^2}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= \sqrt{(−1 − (−7))^2 + (27 − 7)^2}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= \sqrt{6^2 + 20^2}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= \sqrt{436}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= 20,881\kern2mm\ } $$                     [ Q 5.5 ]

  
Answers  6
    
   6.1  y = a(x + p)2 + q
          D(−1,25 ; −15,125) is the turning point
          and thus p = 1,25 and q = -15,125
$$ \hspace*{9 mm}\mathrm{f(x) = a(x + p)^2 + q\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{f(x) = a(x + p)^2 + q\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{f(x) = a(x + 1,25)^2 − 15,125\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{At\ P(-6;30)\ :\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{a((-6) + 1,25)^2 - 15,125 = 30\kern2mm\ } $$
$$ \hspace*{35 mm}\mathrm{22,5625a = 45,125\kern2mm\ } $$
$$ \hspace*{48 mm}\mathrm{a = 2\kern2mm\ } $$
                                                                    [ Q 6.1 ]
    
$$ \hspace*{2 mm}\mathrm{6.2\kern3mmf(x) = 2(x + 1,25)^2 − 15,125\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{= 2(x^2 + 2,5x + 6,25) − 15,125\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{= 2x^2 + 5x − 12\kern2mm\ } $$
          C is the Y-intercept and thus
          C is the point (0 ; −12)
          A and B are the X-intercepts and thus
$$ \hspace*{15 mm}\mathrm{2x^2 + 5x − 12 = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{(2x - 3)(x + 4) = 0\kern2mm\ } $$
                  x = 1,5 or x = −4
          A is the point(−4 ; 0) and B
          the point (1,5 ; 0)
                                                                    [ Q 6.2 ]
    
6.3   M and N are points on the axis of simmetry,
        and thus xN = -1,25.
$$ \hspace*{12 mm}\mathrm{d(MN) = y_{N} − y_{M} = 30\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{y_{N} − 0 = 30\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{y_{N} = 30\kern2mm\ } $$
        N is the point (_1,25 ; 30)               [ Q 6.3 ]
    
6.4   QN = PN
               = yN − yP
               = −1,25 − (−6)
               = 4,75
        Therefore Q is 4,75 units to the right of N
        xQ = −1,5 + 4,75 = 3,5
        yQ = yP = 30
        Q is the point (3,5 ; 30)
                      OR
$$ \hspace*{15 mm}\mathrm{2x^2 + 5x − 12 = 30\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{2x^2 + 5x − 42 = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{(2x - 7)(x + 6) = 0\kern2mm\ } $$
                  x = 3,5 or x = −6
          Q is the point (3,5 ; 30)               [ Q 6.4 ]
    
$$ \hspace*{2 mm}\mathrm{6.5\kern3mmy = mx + c\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{At\ A(-4;0)\ :\ \ m(-4) + c = 0\kern2mm\ } $$
$$ \hspace*{34 mm}\mathrm{-4m + c = 0\ \kern2mm\ (1)\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{At\ Q(3,5;30)\ :\ \ m(3,5) + c = 30\kern2mm\ } $$
$$ \hspace*{38 mm}\mathrm{3,5m + c = 30\ \kern2mm\ (2)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{(2) - (1)\ :\ \ 7,5m = 30\kern2mm\ } $$
$$ \hspace*{49 mm}\mathrm{m = 4\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{Into\ (2)\ :\ \ 4 \times 3,5 + c = 30\kern2mm\ } $$
$$ \hspace*{54 mm}\mathrm{c = 16\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{g(x)\ :\ \ y = 4x + 16\kern2mm\ } $$     [ Q 6.5 ]
    
$$ \hspace*{2 mm}\mathrm{6.6\kern3mmAt\ E\ x = -1,5\kern2mm\ } $$
$$ \hspace*{17 mm}\mathrm{ y = 4(-1,5) + 16\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= 10\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{d(ME) = y_{E} - y_{M}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= 10 - 0\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= 10\kern2mm\ } $$
        ME has a length of 10 units.           [ Q 6.6 ]

  
Answers  7
    
   7.1  y = -x2 + 3x + 10
          C is the Y-intercept and thus
          C is the point (0 ; 10)
          A and B are the X-intercepts, zero points
$$ \hspace*{9 mm}\mathrm{-x^2 + 3x + 10 = 0\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{x^2 - 3x - 10 = 0\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{(x - 5)(x + 2) = 0\kern2mm\ } $$
                  x = 5 or x = −2
      A is the point (-2 ; 0) and B
      the point (5 ; 0)                               [ Q 7.1 ]
    
   7.2  D is the turning point and is therefore
          a point on the axis of simmetry.
$$ \hspace*{9 mm}\mathrm{y = ax^2 + bx + c\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{x = \frac{-b}{2a}-x^2 + 3x + 10 = 0\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{y = -x^2 + 3x + 10\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{x = \frac{-3}{2\times (-1)} = 1,5\kern2mm\ } $$

$$ \hspace*{9 mm}\mathrm{y = -(1,5)^2 + 3(1,5) + 10\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= 12,25\kern2mm\ } $$
          D is the puoit (1,5 ; 12,25).
                          OR
          Write f(x) in the form y = a(x + p)2 + q
          and then write down the coordinates of D.
$$ \hspace*{9 mm}\mathrm{y = -x^2 + 3x + 10\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{= -(x^2 - 3x - 10)\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{= - \Big(x^2 - 3x + \Big(\frac{-3}{2}\Big)^2 - \Big(\frac{-3}{2}\Big)^2 - 10\Big)\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{= - \Big(x - \frac{3}{2}\Big)^2 + \frac{9}{4} + 10\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{= -(x - 1,5)^2 + 12,25\kern2mm\ } $$
          D is the point (1,5 ; 12,25)           [ Q 7.2 ]
    
   7.3  M is the point (1,5 ; 0)                 [ Q 7.3 ]
    
   7.4  P is a point on the axis of simmetry
          and thus P is the point (1,5 ; p).
          Now do the calculation.
$$ \hspace*{33 mm}\mathrm{y = 10 - 2x\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{At\ P(1,5;p)\ :\ p = 10 - 2 \times 1,5\kern2mm\ } $$
$$ \hspace*{35 mm}\mathrm{= 7\kern2mm\ } $$
          P is the point (1,5 ; 7).                    [ Q 7.4 ]
    
$$ \hspace*{2 mm}\mathrm{7.5\kern3mmd(DP) = y_{D} - y_{P}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= 12,25 - 7\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= 5,25\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{d(PM) = y_{P} - y_{M}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= 7 - 0\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= 7\kern2mm\ } $$
          Length of DP = 5,25 units and
          PM is 7 units long.                       [ Q 7.5 ]
    
$$ \hspace*{2 mm}\mathrm{7.6.1\kern3mmd(QS) = y_{Q} - y_{S}\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{= (-x^2 + 3x + 10) - (10 - 2x)\kern2mm\ } $$
$$ \hspace*{23 mm}\mathrm{= -x^2 + 5x\kern2mm\ } $$              [ Q 7.6.1 ]
    
$$ \hspace*{2 mm}\mathrm{7.6.1\kern3mmd(QS) = -x^2 + 5x = 6\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{-x^2 + 5x - 6 = 0\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{x^2 - 5x + 6 = 0\kern2mm\ } $$
$$ \hspace*{19 mm}\mathrm{(x - 6)(x + 1) = 0\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{x = 6\ \ or \ x = -1\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{x = 6\ \ x > 2\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{At\ Q(6;q)\ q = -6^2 + 3 \times 6 + 10\kern2mm\ } $$
$$ \hspace*{31 mm}\mathrm{= -8\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{At\ S(6;s)\ s = 10 - 2 \times 6\kern2mm\ } $$
$$ \hspace*{30mm}\mathrm{= -2\kern2mm\ } $$
          Q is the point (6 ; -8) and S
          the point (6 ; -2).                           [ V 7.6.2 ]
    
$$ \hspace*{2 mm}\mathrm{7.7\kern3mmf(x) = -(x - 1,5)^2 + 12,25\kern2mm\ } $$
          Translate 1,5 units to the left and
          8 units downwards.
$$ \hspace*{9 mm}\mathrm{\therefore \ h(x) = -(x - 1,5 + 1,5)^2 + (12,25 - 8)\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{= -x^2 + 4,25\kern2mm\ } $$           [ Q 7.7 ]