### Example 1

Find the equation of the quadratic graph.

\begin{align} \displaystyle y &= a(x+2)(x+1) \\ 2 &= a(0+2)(0+1) &\text{substitute }(0,2)\\ 2 &= 2a \\ a &= 1 \\ y &= 1(x+2)(x+1) \\ \therefore y &= x^2+3x+2 \\ \end{align}

### Example 2

Find the equation of the quadratic graph.

\begin{align} \displaystyle y &= a(x+1)^2 \\ 1 &= a(0+1)^2 &\text{substitute }(0,1)\\ a &= 1 \\ y &= 1(x+1)^2 \\ \therefore y &= x^2+2x+1 \\ \end{align}

### Example 3

Find the equation of the quadratic whose graph cuts the $x$-axis at $5$, passes through $(2,5)$ and has axis of symmetry $x=1$.

Two $x$-intercepts are equidistant from the axis of symmetry, thus the other $x$-intercept is $-3$.
\begin{align} \displaystyle y &= a(x-5)(x+3) \\ 5 &= a(2-5)(2+3) &\text{substitute }(2,5)\\ 5 &= -15a \\ a &= -\dfrac{1}{3} \\ y &= -\dfrac{1}{3}(x-5)(x+3) \\ &= -\dfrac{1}{3}(x^2-2x-15) \\ \therefore y &= -\dfrac{1}{3}x^2 + \dfrac{2}{3}x + 5 \end{align}