In Exercises 67–69, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. r(x) = (1/2) |x + 2|

Verified step by step guidance

1. Start with the basic absolute value function \(f(x) = |x|\), which is graphed as a V-shaped graph with its vertex at the origin (0,0).
2. Identify the transformation inside the absolute value: \(x + 2\). This represents a horizontal shift of the graph. Since it is \(x + 2\), shift the graph 2 units to the left.
3. Next, look at the coefficient outside the absolute value, which is \(\frac{1}{2}\). This is a vertical compression by a factor of \(\frac{1}{2}\), meaning the graph will be 'flatter' compared to the original \(f(x) = |x|\) graph.
4. Apply the horizontal shift first: move the vertex from (0,0) to (-2,0).
5. Then apply the vertical compression: multiply all the y-values of the shifted graph by \(\frac{1}{2}\). This will give you the graph of \(r(x) = \frac{1}{2} |x + 2|\).

Key Concepts

Absolute Value Function

The absolute value function, f(x) = |x|, outputs the distance of x from zero on the number line, always producing non-negative values. Its graph is a V-shaped figure with the vertex at the origin (0,0), where the function changes direction. Understanding this basic shape is essential for applying transformations.

Horizontal Shifts

A horizontal shift moves the graph left or right without changing its shape. For r(x) = (1/2)|x + 2|, the '+2' inside the absolute value shifts the graph 2 units to the left. This means the vertex of the graph moves from (0,0) to (-2,0).

Vertical Scaling

Vertical scaling changes the steepness of the graph by multiplying the function by a constant. In r(x) = (1/2)|x + 2|, the factor 1/2 compresses the graph vertically, making it less steep than the original absolute value function. This affects the slope of the lines forming the V shape.