https://file.aiquickdraw.com/mathbot/file/1728272212443qf3v2lo6.jpg

You have shared a parametric equation problem. The equations are:

\[
x = 6 - t^2
\]
\[
y = \frac{t}{2}
\]

with the parameter \( t \) in the domain \(-2 \leq t \leq 4\).

### Steps to graph the parametric equations:
1. **Calculate key points** by substituting different values of \( t \) in the given range \([-2, 4]\).
2. **Plot the points** \( (x(t), y(t)) \).
3. **Join the points smoothly** to see the shape of the curve.

Let's calculate the values of \( x \) and \( y \) at key points of \( t \):

For \( t = -2 \):
\[
x = 6 - (-2)^2 = 6 - 4 = 2, \quad y = \frac{-2}{2} = -1
\]

For \( t = -1 \):
\[
x = 6 - (-1)^2 = 6 - 1 = 5, \quad y = \frac{-1}{2} = -0.5
\]

For \( t = 0 \):
\[
x = 6 - 0^2 = 6, \quad y = \frac{0}{2} = 0
\]

For \( t = 1 \):
\[
x = 6 - 1^2 = 6 - 1 = 5, \quad y = \frac{1}{2} = 0.5
\]

For \( t = 2 \):
\[
x = 6 - 2^2 = 6 - 4 = 2, \quad y = \frac{2}{2} = 1
\]

For \( t = 3 \):
\[
x = 6 - 3^2 = 6 - 9 = -3, \quad y = \frac{3}{2} = 1.5
\]

For \( t = 4 \):
\[
x = 6 - 4^2 = 6 - 16 = -10, \quad y = \frac{4}{2} = 2
\]

These points will give a curve when plotted on a coordinate plane.

Would you like a detailed graph of these parametric equations or have any further questions?

Here are 5 related questions you can explore:
1. What is the geometric shape formed by these parametric equations?
2. How would changing the domain of \( t \) affect the graph?
3. Can you express \( y \) as a function of \( x \) by eliminating the parameter \( t \)?
4. How does the curve behave as \( t \to \infty \) or \( t \to -\infty \)?
5. How do parametric equations differ from regular Cartesian equations?

**Tip:** When dealing with parametric equations, it’s helpful to choose symmetric values of \( t \) (like negative and positive values) to better understand the curve's shape.

Graph the following parametric equations: x = 6 - t^2, y = t/2 for the domain -2 <= t <= 4.

Learn how to graph parametric equations x = 6 - t^2 and y = t/2 for the domain -2 <= t <= 4. Understand key points and visualize the curve. Suitable for high school students studying advanced algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation