Question

and identify the graph of the parametric curve.
in exercises 11–26, use an algebraic method to eliminate the parameter

1. ( x = t^2, y = t + 1 ) hint: eliminate ( t ) and solve for ( x ) in terms of ( y ).

Explanation:

Step1: Solve for t from the y - equation

Given \( y = t + 1 \), we can solve for \( t \) by subtracting 1 from both sides. So, \( t=y - 1 \).

Step2: Substitute t into the x - equation

We know that \( x=t^{2} \) and we found that \( t = y - 1 \). Substitute \( t=y - 1 \) into the equation for \( x \). We get \( x=(y - 1)^{2} \). This is a parabola that opens to the right (since the coefficient of \( (y - k)^{2} \) is positive and it is in terms of \( x \) as a function of \( y \)).

Answer:

The equation after eliminating the parameter is \( x=(y - 1)^{2} \), and the graph is a parabola opening to the right with vertex at \( (0,1) \).