Expressing an equation in Cartesian form means rewriting it in terms of $x$ and $y$ only, eliminating any parametric or polar components.

• If the original equation is given in parametric form (e.g., $x = f(t)$, $y = g(t)$), then expressing it in Cartesian form usually involves eliminating the parameter $t$ to obtain a relation between $x$ and $y$.
• This relation can be written in different forms, such as $y = f(x)$ (explicit form) or as an implicit equation $F(x, y) = 0$.
• You do not necessarily need to solve for $x$ explicitly (i.e., $x = ...$). Often, Cartesian form is expressed as $y = f(x)$, but implicit forms like $x^2 + y^2 = 1$ (a circle) are also valid.

Example 1 (Explicit $y = f(x)$):

If you have parametric equations:
$x = t + 1, \quad y = t^2$ Solve for $t$ from $x = t + 1$:
$t = x - 1$ Substituting in $y = t^2$:
$y = (x - 1)^2$ This is the Cartesian equation in the form $y = f(x)$.

Example 2 (Implicit $F(x, y) = 0$):

If you have parametric equations:
$x = \cos t, \quad y = \sin t$ Using the identity $\cos^2 t + \sin^2 t = 1$, we obtain:
$x^2 + y^2 = 1$ This is also a valid Cartesian form, even though it is not explicitly solved for $y$.

Conclusion:

Yes, you can express it as $y = f(x)$, but Cartesian form does not necessarily require solving for $x$ or $y$ explicitly. Implicit forms like $x^2 + y^2 = 1$ are also acceptable.

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Would you like further clarification?

Here are 5 related questions to expand your understanding:

1. How do you convert polar equations to Cartesian form?
2. What is the difference between implicit and explicit Cartesian equations?
3. Can every parametric equation be converted into a Cartesian form?
4. How do you handle cases where eliminating the parameter is difficult?
5. How do you determine if a Cartesian equation represents a function?

Tip: If solving for $y$ leads to multiple values for a single $x$, the equation does not represent a function, but it is still a valid Cartesian equation!