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What's the full equation for this graph?

What's the full equation for this graph?-example-1
User SpaceFozzy
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1 Answer

2 votes
1.
Left side of a graph is a line that passes through points
A=(-1,-1) and
B=(1,4) so we can write its equation as:


(y-y_A)=(y_B-y_A)/(x_B-x_A)(x-x_A)\\\\\\ \big(y-(-1)\big)=(4-(-1))/(1-(-1))\big(x-(-1)\big)\\\\\\ y+1=(4+1)/(1+1)(x+1)\\\\\\y+1=(5)/(2)(x+1)\\\\\\y+1=(5)/(2)x+(5)/(2)\\\\\\y=(5)/(2)x+(5)/(2)-1\\\\\\\boxed{y=(5)/(2)x+(3)/(2)}

2.
Right side of a graph is a parabola with vertex
V=(3,-3) that passes through point
P=(1,1). Its equation:


y=a(x-x_V)^2+y_V\\\\y=a(x-3)^2+(-3)\\\\y=a(x-3)^2-3

Substitute
x=1 and
y=1 to calculate parameter
a


1=a(1-3)^2-3\\\\1=a(-2)^2-3\\\\1=4a-3\\\\4a=4\quad|:4\\\\a=1

and we have the equation of parabola:


y=a(x-3)^2-3\\\\y=1\cdot(x-3)^2-3\\\\y=(x-3)^2-3\\\\y=x^2-2\cdot x\cdot3+3^2-3\\\\y=x^2-6x+9-3\\\\\boxed{y=x^2-6x+6}

From 1. and 2. full equation for graph is:


f(x)=\begin{cases}(5)/(2)x+(3)/(2)\qquad\text{for}\,\,\,x\ \textless \ 1\\\\x^2-6x+6\qquad\text{for}\,\,\,x\geq1\end{cases}
User Mehdi Faraji
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