To find the range of the function $f(x) = |x + 4| + 2$ using graphing technology, we can analyze the function visually and mathematically.

Function Analysis:

1. The absolute value function $|x + 4|$ shifts the graph of $|x|$ to the left by 4 units.
2. Adding 2 to this function shifts the entire graph upward by 2 units.

The vertex of the function occurs where the expression inside the absolute value is zero. This happens when: $x + 4 = 0 \implies x = -4$ At this point, $f(-4) = |(-4) + 4| + 2 = 0 + 2 = 2$. Therefore, the minimum value of $f(x)$ is 2.

Behavior of the Function:

• Since the absolute value function produces non-negative values, $|x + 4| \geq 0$ for all $x$, and after adding 2, $f(x) \geq 2$.
• The function increases without bound as $x$ moves away from $-4$ in either direction.

Range:

Thus, the range of $f(x) = |x + 4| + 2$ is: $\boxed{[2, \infty)}$ This means the function takes values starting from 2 and goes up to infinity.

Would you like me to provide additional details or answer any questions?

5 Related Questions:

1. How do you find the domain of an absolute value function?
2. What is the significance of the vertex in absolute value functions?
3. How does shifting a function horizontally affect its graph?
4. What is the general form of an absolute value function, and how do transformations apply?
5. How can we express the range of more complex absolute value functions?

Tip:

Always check the vertex of absolute value functions as they often determine the minimum or maximum of the function and affect the range.