Calculus Examples

$x = t - 3$ , $y = t^{2} + 5$

Step 1

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

$x = t - 3$

Step 2

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

$t - 3 = x$

Step 3

Add $3$ to both sides of the equation.

$t = x + 3$

Step 4

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

$y = {(x + 3)}^{2} + 5$

Step 5

Simplify ${(x + 3)}^{2} + 5$ .

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

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

$y = (x + 3) (x + 3) + 5$

Step 5.1.2

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

Tap for more steps...

Step 5.1.2.1

Apply the distributive property.

$y = x (x + 3) + 3 (x + 3) + 5$

Step 5.1.2.2

Apply the distributive property.

$y = x \cdot x + x \cdot 3 + 3 (x + 3) + 5$

Step 5.1.2.3

Apply the distributive property.

$y = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 + 5$

Step 5.1.3

Simplify and combine like terms.

Tap for more steps...

Step 5.1.3.1

Simplify each term.

Tap for more steps...

Step 5.1.3.1.1

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 + 5$

Step 5.1.3.1.2

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

$y = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 + 5$

Step 5.1.3.1.3

Multiply $3$ by $3$ .

$y = x^{2} + 3 x + 3 x + 9 + 5$

Step 5.1.3.2

Add $3 x$ and $3 x$ .

$y = x^{2} + 6 x + 9 + 5$

Step 5.2

Add $9$ and $5$ .

$y = x^{2} + 6 x + 14$