We are tasked with graphing the relation represented by the parametric equations:

$x = 2t$ $y = t + 1$ for the parameter $t$ in the range $-2 < t < 2$.

Step 1: Understanding the Parametric Equations

In a parametric equation, $x$ and $y$ are both expressed in terms of a third variable, $t$, which represents the parameter.

• $x = 2t$
• $y = t + 1$

We need to eliminate the parameter $t$ to obtain a Cartesian equation relating $x$ and $y$.

Step 2: Eliminate the Parameter

To eliminate $t$, solve for $t$ in terms of $x$ from the first equation: $x = 2t \implies t = \frac{x}{2}$

Substitute this expression for $t$ into the second equation: $y = t + 1 = \frac{x}{2} + 1$

Thus, the Cartesian equation is: $y = \frac{x}{2} + 1$

This is the equation of a straight line with slope $\frac{1}{2}$ and y-intercept 1.

Step 3: Set Bounds for $t$

The parameter $t$ is restricted by the inequality $-2 < t < 2$. Substituting these bounds into the equation for $x$, we find the corresponding limits for $x$:

$x = 2(-2) = -4 \quad \text{and} \quad x = 2(2) = 4$

Thus, $x$ varies from $-4$ to $4$.

Step 4: Find Points to Plot

Now, we can create a table of values by substituting various values of $t$ within the given range into the parametric equations.

$t$	$x = 2t$	$y = t + 1$
$-2$	$-4$	$-1$
$-1$	$-2$	$0$
$0$	$0$	$1$
$1$	$2$	$2$
$2$	$4$	$3$

These points are: $(-4, -1), (-2, 0), (0, 1), (2, 2), (4, 3)$.

Step 5: Graph the Relation

1. Plot the points $(-4, -1), (-2, 0), (0, 1), (2, 2), (4, 3)$.
2. Draw a straight line passing through these points since the equation $y = \frac{x}{2} + 1$ is a straight line.
3. Ensure that the line only extends from $x = -4$ to $x = 4$ as determined by the bounds on $t$.

The graph will be a segment of the line $y = \frac{x}{2} + 1$ between $x = -4$ and $x = 4$.

Conclusion

The graph represents a straight line segment from the point $(-4, -1)$ to $(4, 3)$. The slope of the line is $\frac{1}{2}$, and the y-intercept is 1.

Do you want further details or explanations on any part of this solution? Here are some related questions to expand your understanding:

1. What is the significance of the slope in the line equation $y = \frac{x}{2} + 1$?
2. How would the graph change if the range of $t$ was $-3 < t < 3$?
3. What is the effect of changing the parametric equation for $x$?
4. How can we convert other parametric equations into Cartesian form?
5. How can you check if a given set of parametric equations represents a straight line?

Tip: When graphing parametric equations, always check if you can convert them to Cartesian form to gain a better understanding of the shape of the graph.