Graph each function. See Examples 1 and 2. ƒ(x)=-3|x|

Verified step by step guidance

1. Step 1: Understand the function. The function given is ƒ(x)=-3|x|. This is a transformation of the basic absolute value function, |x|. The -3 in front of the absolute value symbol means that the graph of the function will be vertically stretched by a factor of 3 and reflected over the x-axis.
2. Step 2: Identify key points. The basic absolute value function, |x|, has its vertex (the lowest point on the graph) at the origin, (0,0). Since the function is ƒ(x)=-3|x|, the vertex will still be at the origin. However, other points on the graph will be affected by the -3. For example, the point (1,1) on the graph of |x| will become (1,-3) on the graph of ƒ(x)=-3|x|.
3. Step 3: Draw the graph. Start by plotting the vertex at the origin. Then plot the point (1,-3) and its reflection across the y-axis, (-1,-3). Because the absolute value function produces a V-shaped graph, you can draw a straight line through these points to complete the graph.
4. Step 4: Check your graph. The graph should be a V shape that opens downward, with the vertex at the origin. The graph should pass through the points (1,-3) and (-1,-3).
5. Step 5: Label your graph. Make sure to label the x and y axes, as well as the vertex and a few other key points on the graph. This will help others understand your graph.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, measures the distance of a number from zero on the number line, always yielding a non-negative result. For example, |3| equals 3, and |-3| also equals 3. This function is crucial for understanding how the graph of ƒ(x) = -3|x| behaves, as it reflects the input values across the x-axis.

Transformation of Functions

Transformations involve altering the basic shape of a function's graph through shifts, stretches, or reflections. In the case of ƒ(x) = -3|x|, the negative sign indicates a reflection over the x-axis, while the coefficient -3 indicates a vertical stretch by a factor of 3. Understanding these transformations helps in accurately sketching the graph.

Graphing Techniques

Graphing techniques involve plotting points and understanding the behavior of functions to create accurate visual representations. For ƒ(x) = -3|x|, one can start by plotting key points, such as (0,0), (1,-3), and (-1,-3), and then use the properties of the absolute value function to complete the graph. Mastery of these techniques is essential for effectively visualizing functions.