Question

use translations to graph the given function.
$t(x)=\sqrt3{x + 2}$

Explanation:

Step1: Identify parent function

Parent function: $f(x)=\sqrt[3]{x}$

Step2: Determine translation type

$t(x)=\sqrt[3]{x+2}$ is $f(x+2)$, so shift left 2 units.

Step3: Plot parent key points

Key points for $f(x)=\sqrt[3]{x}$:
$(-8, -2), (-1, -1), (0, 0), (1, 1), (8, 2)$

Step4: Apply horizontal shift

Subtract 2 from each x-coordinate:
$(-8-2, -2)=(-10, -2)$
$(-1-2, -1)=(-3, -1)$
$(0-2, 0)=(-2, 0)$
$(1-2, 1)=(-1, 1)$
$(8-2, 2)=(6, 2)$

Step5: Graph translated points

Plot $(-10, -2), (-3, -1), (-2, 0), (-1, 1), (6, 2)$ and draw a smooth curve through them.

Answer:

The graph of $t(x)=\sqrt[3]{x+2}$ is the graph of the parent cube root function shifted 2 units to the left, passing through the points $(-10, -2), (-3, -1), (-2, 0), (-1, 1), (6, 2)$.