To obtain the graph of $f(x) = 2|x+7| + 8$ from the graph of $g(x) = |x|$, the following transformations occur:

1. Shifted horizontally by 7 units to the left:

   • This corresponds to the $+7$ inside the absolute value. Shifting left by 7 units moves the vertex from $(0, 0)$ to $(-7, 0)$.

2. Shifted vertically by 8 units upwards:

   • The $+8$ outside the absolute value causes the graph to shift 8 units up, moving the vertex from $(-7, 0)$ to $(-7, 8)$.

3. Vertically stretched by a factor of 2:

   • The coefficient 2 in front of the absolute value causes a vertical stretch by a factor of 2, making the graph steeper compared to the original $g(x) = |x|$.

Location and Direction of the Graph:

• Location of the Vertex: The vertex of the transformed graph is at $(-7, 8)$.
• Quadrant: Since the vertex is at $(-7, 8)$, the vertex is located in the second quadrant of the coordinate plane.
• Direction: The graph opens upwards since the coefficient in front of the absolute value is positive.

Summary:

• Shifted horizontally by 7 units to the left.
• Shifted vertically by 8 units upwards.
• Vertically stretched by a factor of 2.

Would you like more details or have any questions?

Follow-Up Questions:

1. How does changing the coefficient from 2 to -2 affect the graph of $f(x)$?
2. What is the impact on the graph if the vertical shift is changed to a downward shift by 8 units?
3. How would the graph change if the horizontal shift was to the right instead of to the left?
4. What would be the new vertex if the function was $f(x) = 3|x-4| + 6$?
5. How does the transformation affect the width of the graph compared to $g(x) = |x|$?

Tip:

When analyzing graph transformations, remember that horizontal shifts inside the function affect the x-axis, and vertical transformations outside the function affect the y-axis.