Algebra-2

1. Horizontal Shift

We focus on the expression (x + 4):

In this expression, x + 4 indicates a shift in the x-direction.

When we see (x + h) (where h = 4), it indicates a shift to the left by h units.

Conversely, if it were (x - h), it would indicate a shift to the right by h units.

Thus, from (x + 4), we can conclude that the graph shifts left by 4 units.

2. Vertical Reflection and Stretch

Next, we look at the coefficient -2:

The negative sign indicates that the graph will be reflected across the x-axis. This means that all y-values are inverted.

The absolute value (| -2 | = 2) means that the graph will undergo a vertical stretch by a factor of 2.

When the coefficient is greater than 1 (as in this case, with 2), the graph is stretched; if the coefficient were between 0 and 1 (e.g., 0.5), it would lead to a vertical compression.

3. Vertical Shift

Finally, we examine the constant term +5:

The +5 indicates a shift in the y-direction.

A positive constant term indicates a shift up by that number of units.

So, the graph is shifted up by 5 units.

Summary of Transformations

Combining these three points, we can summarize:

Horizontal Shift: Shifted to the left by 4 units.

Vertical Reflection and Stretch: The graph is reflected across the x-axis and stretched vertically by a factor of 2.

Vertical Shift: Shifted up by 5 units.

Conclusion

The complete transformations of the function

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