Use the graph of y = f(x) to graph each function g.

g(x) = -f(x) +3

Verified step by step guidance

1. Step 1: Analyze the given graph of y = f(x). The graph is a horizontal line segment from (1, -3) to (4, -3). This means that f(x) = -3 for all x in the interval [1, 4].
2. Step 2: Understand the transformation g(x) = -f(x) + 3. The transformation involves two steps: (a) Negate the values of f(x), which means flipping the graph vertically, and (b) Add 3 to the result, which shifts the graph upward by 3 units.
3. Step 3: Apply the negation transformation. Since f(x) = -3, negating it gives -f(x) = 3. The graph is now a horizontal line segment at y = 3 for x in [1, 4].
4. Step 4: Apply the vertical shift transformation. Adding 3 to -f(x) gives g(x) = 3 + 3 = 6. The graph is now a horizontal line segment at y = 6 for x in [1, 4].
5. Step 5: Plot the transformed graph of g(x). The new graph is a horizontal line segment from (1, 6) to (4, 6).

Key Concepts

Function Transformation

Function transformation refers to the changes made to the graph of a function based on modifications to its equation. In this case, the function g(x) = -f(x) + 3 involves a vertical shift and a reflection. The negative sign indicates a reflection over the x-axis, while the '+3' shifts the graph upward by three units.

Reflection Over the X-Axis

Reflecting a function over the x-axis means that for every point (x, y) on the original graph, the corresponding point on the reflected graph will be (x, -y). This transformation changes the sign of the output values, effectively flipping the graph upside down. For the function g(x) = -f(x), this reflection alters the horizontal line segment of f(x) from y = -3 to y = 3.

Vertical Shift

A vertical shift occurs when a constant is added to or subtracted from a function's output. In the function g(x) = -f(x) + 3, the '+3' indicates that the entire graph of -f(x) is moved up by three units. This means that after reflecting the original function, the new graph will be positioned higher on the y-axis, resulting in a horizontal line segment at y = 0.