Precalculus Examples

$x (t) = 3 + t$ , $y (t) = - t^{2}$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = 3 + t$

Step 2

Rewrite the equation as $3 + t = x$ .

$3 + t = x$

Step 3

Subtract $3$ from both sides of the equation.

$t = x - 3$

Step 4

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = - {(x - 3)}^{2}$

Step 5

Simplify $- {(x - 3)}^{2}$ .

Tap for more steps...

Step 5.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$y = - ((x - 3) (x - 3))$

Step 5.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Tap for more steps...

Step 5.2.1

Apply the distributive property.

$y = - (x (x - 3) - 3 (x - 3))$

Step 5.2.2

Apply the distributive property.

$y = - (x \cdot x + x \cdot - 3 - 3 (x - 3))$

Step 5.2.3

Apply the distributive property.

$y = - (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Step 5.3

Simplify and combine like terms.

Tap for more steps...

Step 5.3.1

Simplify each term.

Tap for more steps...

Step 5.3.1.1

Multiply $x$ by $x$ .

$y = - (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Step 5.3.1.2

Move $- 3$ to the left of $x$ .

$y = - (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3)$

Step 5.3.1.3

Multiply $- 3$ by $- 3$ .

$y = - (x^{2} - 3 x - 3 x + 9)$

Step 5.3.2

Subtract $3 x$ from $- 3 x$ .

$y = - (x^{2} - 6 x + 9)$

Step 5.4

Apply the distributive property.

$y = - x^{2} - (- 6 x) - 1 \cdot 9$

Step 5.5

Simplify.

Tap for more steps...

Step 5.5.1

Multiply $- 6$ by $- 1$ .

$y = - x^{2} + 6 x - 1 \cdot 9$

Step 5.5.2

Multiply $- 1$ by $9$ .

$y = - x^{2} + 6 x - 9$