Question

find parametric equations that define the curve starting at $(1,0)$ and ending at $(6,9)$ as shown. let parameter $t$ start at 0 and end at 9. $y = t$ (complete the equation, with the right simplest possible form.) $x = square$ (complete the equation, with the right simplest possible form.)

Explanation:

Step1: Determine the relationship between \(x\) and \(t\)

We know that when \(t = 0\), \(x=1\) and when \(t = 9\), \(x = 6\).
We use the formula for a linear relationship \(x=at + b\).
Substitute \(t = 0,x = 1\) into \(x=at + b\), we get \(1=a\times0 + b\), so \(b = 1\).
Substitute \(t = 9,x = 6,b = 1\) into \(x=at + b\), we have \(6=a\times9+1\).
Solve the equation \(6 = 9a+1\) for \(a\):

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So \(x=\frac{5}{9}t+1\)

Answer:

\(x=\frac{5}{9}t + 1\)