Question

use transformations to graph the function.
$f(x) = -3|x + 2| - 4$

Explanation:

Step1: Identify parent function

Parent function: $g(x)=|x|$, vertex at $(0,0)$

Step2: Apply horizontal shift

Shift left 2 units: $g(x+2)=|x+2|$, vertex at $(-2,0)$

Step3: Apply vertical stretch/reflection

Stretch by 3, reflect over x-axis: $-3|x+2|$, vertex remains $(-2,0)$

Step4: Apply vertical shift

Shift down 4 units: $f(x)=-3|x+2|-4$, vertex at $(-2,-4)$

Step5: Plot key points

For $x=-2$: $f(-2)=-3|0|-4=-4$
For $x=-1$: $f(-1)=-3|1|-4=-7$
For $x=-3$: $f(-3)=-3|1|-4=-7$
For $x=0$: $f(0)=-3|2|-4=-10$
For $x=-4$: $f(-4)=-3|2|-4=-10$

Answer:

The graph of $f(x)=-3|x+2|-4$ is a downward-opening V-shape with vertex at $(-2, -4)$, passing through points like $(-1, -7)$, $(-3, -7)$, $(0, -10)$, $(-4, -10)$, and extending infinitely in both directions with slopes of 3 and -3 on its right and left branches respectively.