\begin{align} y & = (x + 1)(x + 3) \\ y & = x^2 + 3x + x + 3 \\ y & = x^2 + 4x + 3 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (0 + 1)(0 + 3) \\ y & = 3 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x + 1)(x + 3) \\ \\ x & = -1 \text{ or } x = - 3 \phantom{000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {-1 + (-3) \over 2} \\ x & = -2 \\ \\ \text{Let } & x = -2, \\ y & = (-2 + 1)(-2 + 3) \\ y & = -1 \\ \\ \text{Minimum point: } & (-2, -1) \\ \end{align}

\begin{align} y & = (x - 2)(x + 4) \\ y & = x^2 + 4x - 2x - 8 \\ y & = x^2 + 2x - 8 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (0 - 2)(0 + 4) \\ y & = -8 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x - 2)(x + 4) \\ \\ x & = 2 \text{ or } x = -4 \phantom{000000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {2 + (-4) \over 2} \\ x & = -1 \\ \\ \text{Let } & x = -1, \\ y & = (-1 -2)(-1 + 4) \\ y & = -9 \\ \\ \text{Minimum point: } & (-1, -9) \end{align}

\begin{align} y & = -(x + 1)(x - 5) \\ y & = -(x^2 - 5x + x - 5) \\ y & = -(x^2 - 4x - 5) \\ y & = - x^2 + 4x + 5 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ \\ \text{Let } & x = 0, \\ y & = -(0 + 1)(0 - 5) \\ y & = 5 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = -(x + 1)(x - 5) \\ \\ x & = -1 \text{ or } x = 5 \phantom{00000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {-1 + 5 \over 2} \\ x & = 2 \\ \\ \text{Let } & x = 2, \\ y & = -(2 + 1)(2 - 5) \\ y & = 9 \\ \\ \text{Maximum point: } & (2, 9) \end{align}

\begin{align} y & = -(x - 1)(x + 6) \\ y & = -(x^2 + 6x - x - 6) \\ y & = -(x^2 + 5x - 6) \\ y & = - x^2 - 5x + 6 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ \\ \text{Let } & x = 0, \\ y & = -(0 - 1)(0 + 6) \\ y & = 6 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = -(x - 1)(x + 6) \\ \\ x & = 1 \text{ or } x = -6 \phantom{00000000000} [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{Maximum point: } & (-2.5, 12.25) \end{align}

\begin{align} y & = (3 - x)(x + 2) \\ y & = 3x + 6 - x^2 - 2x \\ y & = - x^2 + x + 6 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ \\ \text{Let } & x = 0, \\ y & = (3 - 0)(0 + 2) \\ y & = 6 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (3 - x)(x + 2) \\ \\ x & = 3 \text{ or } x = -2 \phantom{00000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {3 + (-2) \over 2} \\ x & = 0.5 \\ \\ \text{Let } & x = 0.5, \\ y & = (3 - 0.5)(0.5 + 2) \\ y & = 6.25 \\ \\ \text{Maximum point: } & (0.5, 6.25) \end{align}

\begin{align} y & = (2 - x)(4 - x) \\ y & = 8 - 2x - 4x + x^2 \\ y & = x^2 - 6x + 8 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (2 - 0)(4 - 0) \\ y & = 8 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (2 - x)(4 - x) \\ \\ x & = 2 \text{ or } x = 4 \phantom{000000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {2 + 4 \over 2} \\ x & = 3 \\ \\ \text{Let } & x = 3, \\ y & = (2 - 3)(4 - 3) \\ y & = -1 \\ \\ \text{Minimum point: } & (3, -1) \end{align}

\begin{align} y & = x^2 - 4 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = (x - 0)^2 - 4 \\ \\ \text{Minimum point: } & (0, -4) \\ \\ \text{Line of symmetry: } & x = 0 \\ \\ \text{Let } & x = 0, \\ y & = (0)^2 - 4 \\ y & = - 4 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = -x^2 + 6 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ y & = -(x - 0)^2 + 6 \\ \\ \text{Maximum point: } & (0, 6) \\ \\ \text{Line of symmetry: } & x = 0 \\ \\ \text{Let } & x = 0, \\ y & = -(0)^2 + 6 \\ y & = 6 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = (x - 3)^2 - 2 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Minimum point: } & (3, -2) \\ \\ \text{Line of symmetry: } & x = 3 \\ \\ \text{Let } & x = 0, \\ y & = (0 - 3)^2 - 2 \\ y & = 7 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = (x + 1)^2 - 3 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = [x - (-1)]^2 - 3 \\ \\ \text{Minimum point: } & (-1, -3) \\ \\ \text{Line of symmetry: } & x = -1 \\ \\ \text{Let } & x = 0, \\ y & = (0 + 1)^2 - 3 \\ y & = -2 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = -(x + 2)^2 + 3 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ y & = -[x - (-2)]^2 + 3 \\ \\ \text{Maximum point: } & (-2, 3) \\ \\ \text{Line of symmetry: } & x = -2 \\ \\ \text{Let } & x = 0, \\ y & = -(0 + 2)^2 + 3 \\ y & = -1 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = -(x - 4)^2 + 5 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ \\ \text{Maximum point: } & (4, 5) \\ \\ \text{Line of symmetry: } & x = 4 \\ \\ \text{Let } & x = 0, \\ y & = - (0 - 4)^2 + 5 \\ y & = -11 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = x^2 + {3 \over 4}x \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = x \left(x + {3 \over 4}\right) \\ \\ \text{Let } & x = 0, \\ y & = (0)\left(0 + {3 \over 4}\right) \\ y & = 0 \phantom{000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = x \left(x + {3 \over 4}\right) \\ \\ x & = 0 \text{ or } x = -{3 \over 4} \phantom{000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = { 0 + \left(-{3 \over 4}\right) \over 2} \\ x & = -{3 \over 8} \\ \\ \text{Let } & x = -{3 \over 8}, \\ y & = \left(-{3 \over 8}\right) \left(-{3 \over 8} + {3 \over 4}\right) \\ y & = -{9 \over 64} \\ \\ \text{Minimum point: } & \left(-{3 \over 8}, -{9 \over 64}\right) \end{align}

\begin{align} y & = -(x^2 - x) \\ y & = -x^2 + x \phantom{0000000} [\text{Maximum curve } (\cap)] \\ \\ \text{Let } & x = 0, \\ y & = -(0)^2 + (0) \\ y & = 0 \phantom{000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = -x^2 + x \\ 0 & = -x(x - 1) \\ \\ x & = 0 \text{ or } x = 1 \phantom{00000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = { 0 + 1 \over 2} \\ x & = 0.5 \\ \\ \text{Let } & x = 0.5, \\ y & = -(0.5)^2 + (0.5) \\ y & = 0.25 \\ \\ \text{Maximum point: } & (0.5, 0.25) \end{align}

\begin{align} y & = x^2 + x - 6 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (0)^2 + (0) - 6 \\ y & = -6 \phantom{000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = x^2 + x - 6 \\ 0 & = (x - 2)(x + 3) \phantom{000000000} [\text{Use result from (i)}] \\ \\ x & = 2 \text{ or } x = -3 \phantom{00000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = { 2 + (-3) \over 2} \\ x & = -0.5 \\ \\ \text{Let } & x = -0.5, \\ y & = (-0.5)^2 + (-0.5) - 6 \\ y & = -6.25 \\ \\ \text{Minimum point: } & (-0.5, -6.25) \end{align}

\begin{align} y & = x^2 - 4x + 3 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (0)^2 - 4(0) + 3 \\ & = 3 \phantom{000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = x^2 - 4x + 3 \\ 0 & = (x - 1)(x - 3) \\ \\ x & = 1 \text{ or } x = 3 \phantom{00000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = { 1 + 3 \over 2} \\ x & = 2 \\ \\ \text{Let } & x = 2, \\ y & = (2)^2 - 4(2) + 3 \\ y & = -1 \\ \\ \text{Minimum point: } & (2, -1) \end{align}

\begin{align} y & = - x^2 - 7x - 15 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ y & = - (x + 3.5)^2 - 2.75 \\ y & = - [x - (-3.5)]^2 - 2.75 \\ \\ \text{Maximum point: } & (-3.5, -2.75) \\ \\ \text{Let } & x = 0, \\ y & = - (0 + 3.5)^2 - 2.75 \\ y & = -15 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = x^2 - 3x + 4 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = (x - 1.5)^2 + 1.75 \\ \\ \text{Minimum point: } & (1.5, 1.75) \\ \\ \text{Let } & x = 0, \\ y & = (0 - 1.5)^2 + 1.75 \\ y & = 4 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = - x^2 - 10x - 25 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ y & = - (x + 5)^2 \\ y & = - [x - (-5)]^2 + 0 \\ \\ \text{Maximum point: } & (-5, 0) \\ \\ \text{Let } & x = 0, \\ y & = - (0 + 5)^2 \\ y & = -25 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = x^2 - 8x + 16 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = (x - 4)^2 \\ y & = (x - 4)^2 + 0 \\ \\ \text{Minimum point: } & (4, 0) \\ \\ \text{Let } & x = 0, \\ y & = (0 - 4)^2 \\ y & = 16 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = - x^2 + 6x - 6 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ y & = -(x - 3)^2 + 3 \\ \\ \text{Maximum point: } & (3, 3) \\ \\ \text{Let } & x = 0, \\ y & = -(0 - 3)^2 + 3 \\ y & = -6 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y & = x^2 - 8x + 5 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = (x - 4)^2 - 11 \\ \\ \text{Minimum point: } & (4, -11) \\ \\ \text{Let } & x = 0, \\ y & = (0 - 4)^2 - 11 \\ y & = 5 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

\begin{align} y = \left[ x - \left(-{1 \over 2}\right) \right]^2 & + {3 \over 4} \phantom{0000000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Minimum point: } & \left(-{1 \over 2}, {3 \over 4} \right) \\ \\ \text{Let } & x = 0, \\ y & = \left[0 - \left(-{1 \over 2}\right) \right]^2 + {3 \over 4} \\ y & = 1 \phantom{000000000000000000} [ y \text{-intercept}] \end{align}