graph the relation represented by the following parametric equations x=2t  y=t+1  -2<t<2 please show all your work

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.

Learn how to graph parametric equations x = 2t and y = t + 1 for the range -2 < t < 2. Follow the step-by-step solution that includes eliminating the parameter to obtain the Cartesian equation, plotting points, and drawing the line segment for this relation. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation