The function you provided is:

$f(x) = 2|x+3| - 4$

This function consists of an absolute value expression, meaning the output behaves differently depending on whether $x+3$ is positive or negative. Let's break it down:

Case 1: $x+3 \geq 0$ (i.e., $x \geq -3$)

In this case, the absolute value expression $|x+3| = x+3$. The function becomes: $f(x) = 2(x+3) - 4 = 2x + 6 - 4 = 2x + 2$

Case 2: $x+3 < 0$ (i.e., $x < -3$)

Here, $|x+3| = -(x+3)$, so the function becomes: $f(x) = 2(-(x+3)) - 4 = -2(x+3) - 4 = -2x - 6 - 4 = -2x - 10$

Thus, the piecewise form of the function is:

2x + 2 & \text{if } x \geq -3 \\ -2x - 10 & \text{if } x < -3 \end{cases}$$ ### Graphical Interpretation - For $$x \geq -3$$, the function behaves like a line with slope 2. - For $$x < -3$$, the function behaves like a line with slope -2. The point $$x = -3$$ is where the two cases meet, and you can calculate $$f(-3)$$ to confirm this: $$f(-3) = 2|(-3)+3| - 4 = 2(0) - 4 = -4$$ So the point $$(-3, -4)$$ is where the two pieces of the function connect. Would you like to see more details or have any questions? --- Here are 5 related questions to further explore this topic: 1. How do you find the vertex of this absolute value function? 2. What happens to the graph if the coefficient of the absolute value is changed? 3. How do you determine the slope of the lines in each case? 4. What is the domain and range of this function? 5. Can you find the x-intercepts and y-intercepts of this function? **Tip:** For absolute value functions, always consider the piecewise nature of the function and analyze it for different intervals based on when the expression inside the absolute value becomes positive or negative.