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