To graph the function $f(x) = -4|x|$, we can analyze it step by step using transformations and the definition of absolute value.

Step 1: Understand the Parent Function

The parent function here is $|x|$, which is the absolute value function. The graph of $y = |x|$ is a "V"-shaped graph, with a vertex at $(0, 0)$, where:

• For $x \geq 0$, $y = x$
• For $x < 0$, $y = -x$

Step 2: Transformation by -4

Now, apply the transformation $-4|x|$. This transformation does two things:

1. Negative sign (-): Reflects the graph of $|x|$ over the x-axis, turning the "V" shape upside down.
2. Vertical stretch by 4: Multiplies the output (y-values) of the function by 4, making the graph steeper.

Thus, the graph of $f(x) = -4|x|$ will have the following characteristics:

• For $x \geq 0$, $f(x) = -4x$
• For $x < 0$, $f(x) = 4x$

Step 3: Key Points on the Graph

• The vertex of the graph remains at $(0, 0)$.
• As $x$ increases from 0, the graph decreases linearly with a slope of -4.
• As $x$ decreases from 0, the graph increases linearly with a slope of 4.

Graph Description

• The graph is V-shaped, opening downward, with a vertex at $(0, 0)$.
• For $x > 0$, the slope is -4 (steep negative slope).
• For $x < 0$, the slope is 4 (steep positive slope).

If you want to visualize it, think of an inverted, narrower "V" due to the vertical stretching and the reflection.

Would you like further details or a visual representation of this graph? Here are some related questions:

1. How does changing the coefficient from -4 to -2 affect the graph?
2. What would happen if the function was $4|x|$ instead of $-4|x|$?
3. How do transformations affect absolute value functions in general?
4. Can you describe the domain and range of this function?
5. How do we compute the slope of the lines for each side of the graph?

Tip: To graph absolute value functions with transformations, focus on the vertex and how reflections and stretches affect the slope and orientation.