In Exercises 33-44, use the graph of y = f(x) to graph each function g. g(x) = -½ ƒ ( x + 2) —2

Verified step by step guidance

1. Identify the transformations applied to the function $f(x)$ to obtain $g(x)=-\frac{1}{2}f(x+2)-2$.
2. Recognize that $f(x+2)$ represents a horizontal shift of the graph of $f(x)$ to the left by 2 units.
3. Understand that $-\frac{1}{2}f(x+2)$ indicates a vertical compression by a factor of $\frac{1}{2}$ and a reflection across the x-axis.
4. Note that the $-2$ at the end of the function represents a vertical shift downward by 2 units.
5. Combine all transformations: shift the graph of $f(x)$ left by 2 units, reflect it across the x-axis, compress it vertically by $\frac{1}{2}$, and then shift it downward by 2 units.

Key Concepts

Function Transformation

Function transformation refers to the changes made to the graph of a function based on modifications to its equation. These transformations include shifts, stretches, compressions, and reflections. In the given function g(x), the transformations involve a horizontal shift, a vertical shift, and a vertical compression, which are essential for accurately graphing the new function based on the original f(x).

Horizontal Shifts

Horizontal shifts occur when the input variable x is altered by adding or subtracting a constant. In the function g(x) = -½ ƒ (x + 2), the term (x + 2) indicates a shift to the left by 2 units. This means that every point on the graph of f(x) will move leftward, affecting the overall position of the graph of g(x).

Vertical Compression and Reflection

Vertical compression and reflection involve scaling the output of a function and flipping it across the x-axis. In g(x), the coefficient -½ indicates a vertical compression by a factor of ½ and a reflection across the x-axis due to the negative sign. This means that the values of f(x) are halved and inverted, which significantly alters the shape and orientation of the graph.