# What is the slope of f(t) = (t-1,t+3) at t =1?

Feb 16, 2016

$1$

#### Explanation:

The derivative of the parametric function

$f \left(t\right) = \left(x \left(t\right) , y \left(t\right)\right)$

is

$f ' \left(t\right) = \frac{x ' \left(t\right)}{y ' \left(t\right)}$

The given parametric function is

$f \left(t\right) = \left(t - 1 , t + 3\right)$

which means that

$x \left(t\right) = t - 1$
$y \left(t\right) = t + 3$

Differentiation of both functions with respect to $t$ yields

$x ' \left(t\right) = 1$
$y ' \left(t\right) = 1$

so the entire parametric function's derivative is

$f ' \left(t\right) = \frac{x ' \left(t\right)}{y ' \left(t\right)} = \frac{1}{1} = 1$

This means that, irrespective of the value of $t$, the slope of the tangent line to the function is always $1$.