The given function is $f(x) = -2|x + 1| + 4$, which involves the absolute value function.

Steps to sketch the graph:

1. Basic Shape of Absolute Value Function:

   • The base function is $|x + 1|$, which is a V-shaped graph centered at $x = -1$, as the vertex of $|x + 1|$ occurs where the expression inside the absolute value is zero ($x = -1$).

2. Reflection and Stretch:

   • The coefficient $-2$ reflects the graph across the x-axis (flipping it upside down) and stretches it vertically by a factor of 2. This makes the "V" sharper and inverted.

3. Vertical Shift:

   • The $+4$ shifts the entire graph upwards by 4 units, moving the vertex of the graph to $(-1, 4)$.

Key Points:

• Vertex: The vertex of the graph is at $(-1, 4)$.
• Slope: The slope of the left side of the graph (for $x < -1$) is $2$, and the slope of the right side (for $x > -1$) is $-2$, due to the negative reflection.

Sketch:

The graph forms a V-shape:

• It starts at $(-1, 4)$.
• To the left of $x = -1$, it decreases with a slope of $2$.
• To the right of $x = -1$, it decreases with a slope of $-2$ (due to reflection).

Would you like to see a detailed plot of the graph, or do you have any further questions?

Related Questions:

1. How do you find the vertex of an absolute value function?
2. What happens to the graph when the coefficient of $|x + 1|$ changes from $2$ to $-2$?
3. How do you determine the domain and range of $f(x) = -2|x + 1| + 4$?
4. What effect does a vertical shift have on the graph of a function?
5. How can you derive the equation of an absolute value graph from its vertex?

Tip: Always start by identifying the vertex of an absolute value function, as it determines the basic shape and position of the graph.