Let's break down the function $f(x) = -2|3x| + 1$ step by step to identify and explain the transformations involved in the graph:

1. Start with the basic function:

The core function is $|x|$, which is the absolute value function. Its graph is a "V" shape with a vertex at $(0, 0)$, opening upwards.

2. Horizontal Stretch/Compression:

The term inside the absolute value is $|3x|$. This represents a horizontal compression by a factor of $\frac{1}{3}$.

• Normally, the graph of $|x|$ stretches equally along the x-axis, but multiplying by 3 inside the absolute value compresses it horizontally by a factor of 3.
• Points on the graph that were originally at $(1,1)$ and $(-1,1)$ on $|x|$ will now be at $\left(\frac{1}{3},1\right)$ and $\left(-\frac{1}{3},1\right)$.

3. Vertical Stretch and Reflection:

The coefficient $-2$ outside of the absolute value function has two effects:

• Vertical stretch by a factor of 2: Multiplying the output by 2 makes the "V" steeper. The slopes of the arms change from $\pm 1$ to $\pm 2$.
• Reflection over the x-axis: The negative sign flips the graph, so the "V" now opens downwards instead of upwards.

4. Vertical Translation:

The "+1" at the end of the function translates the graph upward by 1 unit. This means that the vertex of the graph, which was originally at $(0, 0)$, is now moved to $(0, 1)$.

Summary of Transformations:

1. Horizontal compression by a factor of 3.
2. Vertical stretch by a factor of 2.
3. Reflection over the x-axis (opening downwards).
4. Vertical shift up by 1 unit.

Graphing these transformations:

1. Start with the basic $|x|$ graph.
2. Apply the horizontal compression: compress it by a factor of 3 along the x-axis.
3. Apply the vertical stretch and reflection: stretch it vertically and flip it to open downwards.
4. Finally, shift the entire graph up by 1 unit.

Would you like me to graph this for you or go into more details about any of the transformations?

5 Related Questions:

1. How does the graph change if we replace $-2$ with $-1$?
2. What happens if the vertical shift changes from +1 to -2?
3. How would the transformations differ if we had $|x-2|$ inside the absolute value?
4. What is the effect of multiplying the absolute value by a fraction instead of 2?
5. How can we algebraically verify the vertex and slope changes of the graph?

Tip:

When interpreting transformations, always look for changes in signs or coefficients inside and outside the function for horizontal and vertical shifts or reflections.