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

Verified step by step guidance

1. Identify the base function to graph, which is the absolute value function \(f(x) = |x|\). This function creates a 'V' shape with its vertex at the origin (0,0), opening upwards.
2. Understand the transformation applied to the base function. The given function is \(f(x) = -3|x|\), which means the absolute value function is first multiplied by 3, stretching it vertically by a factor of 3, and then multiplied by -1, reflecting it across the x-axis.
3. Plot the vertex of the graph at the origin (0,0), since the absolute value function's vertex remains unchanged by vertical stretching or reflection.
4. Choose several x-values (both positive and negative), calculate the corresponding y-values using \(f(x) = -3|x|\), and plot these points. For example, for \(x=1\), \(f(1) = -3|1| = -3\), and for \(x=-1\), \(f(-1) = -3|-1| = -3\).
5. Draw the graph by connecting the plotted points with straight lines forming a 'V' shape that opens downward due to the negative sign, and is steeper than the basic \(|x|\) graph because of the factor 3.

Key Concepts

Absolute Value Function

The absolute value function, denoted |x|, outputs the distance of x from zero on the number line, always yielding a non-negative result. Its graph is a V-shaped curve with the vertex at the origin, reflecting all negative inputs to positive outputs.

Vertical Stretch and Reflection

Multiplying a function by a negative constant, like -3, reflects the graph across the x-axis and stretches it vertically by a factor of 3. This changes the shape and orientation of the graph, flipping it upside down and making it steeper.

Graphing Piecewise Functions

Absolute value functions can be expressed as piecewise linear functions, defining different expressions for x ≥ 0 and x < 0. Understanding this helps in plotting the graph accurately by considering each piece separately.