To compare the graph of $g(x) = 12|x + 3| - 1$ to the graph of $f(x) = |x|$, we'll break down the transformations step by step.

1. Starting Function $f(x) = |x|$

• This is the basic absolute value function, which has a "V" shape centered at the origin $(0, 0)$, opening upwards.

2. Horizontal Shift

• The function $g(x) = |x + 3|$ indicates a horizontal shift of $f(x)$ by 3 units to the left. This is because the expression inside the absolute value, $x + 3$, means that every $x$ value is effectively reduced by 3 before applying the absolute value.

3. Vertical Stretch

• The function $g(x) = 12|x + 3|$ introduces a vertical stretch by a factor of 12. This makes the "V" shape of the graph steeper. Every y-value of $|x + 3|$ is multiplied by 12, stretching the graph vertically.

4. Vertical Shift

• Finally, $g(x) = 12|x + 3| - 1$ involves a vertical shift downwards by 1 unit. This means the entire graph is moved down by 1 unit.

Summary of Transformations:

• Start with $f(x) = |x|$.
• Shift the graph 3 units to the left.
• Apply a vertical stretch by a factor of 12.
• Move the graph 1 unit downwards.

Visual Comparison:

• $f(x) = |x|$: "V" shape centered at $(0,0)$.
• $g(x) = 12|x + 3| - 1$: Much steeper "V" shape centered at $(-3,-1)$.

Key Points:

• The vertex of $g(x)$ is at $(-3, -1)$.
• The graph is 12 times steeper than $f(x)$.

Would you like to see the graphs or need any further details?

Here are 5 related questions:

1. How would the graph change if the coefficient 12 were different?
2. What happens to the graph if we shift $f(x) = |x|$ by 3 units to the right instead?
3. How do you determine the vertex of a transformed absolute value function?
4. What effect does multiplying by a negative number have on the graph?
5. How would the graph look if we added a vertical stretch instead of compression?

Tip: Remember, horizontal shifts inside the function's argument $|x + h|$ always move the graph in the opposite direction of the sign.