In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = 2|x+3|

Verified step by step guidance

1. Start by graphing the basic absolute value function, $f(x)=|x|$. This graph is a V-shaped graph with its vertex at the origin (0,0) and opens upwards.
2. Identify the transformation needed for $h(x)=2|x+3|$. Notice that $|x+3|$ indicates a horizontal shift.
3. The expression $|x+3|$ represents a shift of the graph of $|x|$ to the left by 3 units. So, move the vertex of the graph of $|x|$ from (0,0) to (-3,0).
4. The coefficient 2 in front of $|x+3|$ indicates a vertical stretch. This means that each point on the graph of $|x+3|$ is stretched away from the x-axis by a factor of 2.
5. Combine these transformations: shift the graph of $|x|$ left by 3 units and then apply a vertical stretch by a factor of 2 to obtain the graph of $h(x)=2|x+3|$.

Key Concepts

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. This function has a V-shaped graph that opens upwards, with its vertex at the origin (0,0). Understanding this function is crucial as it serves as the foundation for graphing transformations.

Transformations of Functions

Transformations involve altering the graph of a function through shifts, stretches, or reflections. For example, adding a constant inside the absolute value (like +3) shifts the graph horizontally, while multiplying by a constant (like 2) vertically stretches it. Recognizing these transformations helps in accurately graphing new functions based on the original.

Graphing Techniques

Graphing techniques involve plotting points and understanding how transformations affect the shape and position of a graph. For the function h(x) = 2|x+3|, one must first graph f(x) = |x|, then apply the horizontal shift left by 3 units and a vertical stretch by a factor of 2. Mastery of these techniques is essential for visualizing and interpreting functions.