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find the cartesian equation for y in terms of x

The given parametric equations are:

\[
x = 1 - t^2
\]
\[
y = 4 - t
\]

We need to find the Cartesian equation \( y \) in terms of \( x \).

### Step 1: Solve the equation for \( t \) in terms of \( y \)
From the equation \( y = 4 - t \), solve for \( t \):

\[
t = 4 - y
\]

### Step 2: Substitute \( t \) into the equation for \( x \)
Substitute \( t = 4 - y \) into the equation \( x = 1 - t^2 \):

\[
x = 1 - (4 - y)^2
\]

### Step 3: Expand the equation
Expand \( (4 - y)^2 \):

\[
x = 1 - (16 - 8y + y^2)
\]

\[
x = 1 - 16 + 8y - y^2
\]

\[
x = -15 + 8y - y^2
\]

### Step 4: Rearrange the equation
Rearrange to express \( y \) explicitly:

\[
y^2 - 8y + (x + 15) = 0
\]

This is the Cartesian equation for \( y \) in terms of \( x \).

Do you want further details or have any questions?

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**Related Questions:**
1. How can we solve the quadratic equation to express \( y \) explicitly?
2. What does the resulting equation represent geometrically?
3. How do you determine the range of \( y \) given the parametric form?
4. What is the effect of changing the value of \( t \) on the curve?
5. How would you find the vertex of the parabola formed by the equation?
6. Can you graph the equation \( x = 1 - t^2 \) and \( y = 4 - t \)?
7. What is the domain of the Cartesian equation derived?
8. How would the Cartesian equation change if the parameter \( t \) is shifted by a constant?

**Tip:** When dealing with parametric equations, always try to eliminate the parameter step by step and carefully expand and rearrange terms to derive the Cartesian form.

Learn how to find the Cartesian equation for y in terms of x using parametric equations. Follow a detailed step-by-step solution with key algebraic concepts, including parametric equations, Cartesian equations, and quadratic equations. Suitable for students in Grades 10-12.

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