Graph each function.

Verified step by step guidance

1. Identify the piecewise function and its two parts: \(f(x) = |4x|\) for \(x < 4\) and \(f(x) = -x + 20\) for \(x \geq 4\).
2. For the first part, \(f(x) = |4x|\), recognize that the absolute value function outputs the positive value of \$4x\(. So for \)x < 4\(, calculate \)f(x)\( by taking the absolute value of \)4x$.
3. For the second part, \(f(x) = -x + 20\), this is a linear function with slope \(-1\) and y-intercept \$20\(. For \)x \geq 4\(, plot this line starting at \)x=4$.
4. Find the value of the function at the boundary point \(x=4\) for both parts to check continuity: calculate \(|4(4)|\) and \(-4 + 20\) to see if the function values match or if there is a jump.
5. Plot the graph by drawing the curve for \(f(x) = |4x|\) on the left side of \(x=4\) and the line \(f(x) = -x + 20\) on the right side of \(x=4\), using the boundary values to connect or separate the pieces.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions depending on the input value's interval. Understanding how to interpret and graph each piece separately is essential, as the function's rule changes at specified points, called breakpoints.

Absolute Value Function

The absolute value function, |x|, outputs the distance of x from zero, always non-negative. When graphing, it creates a V-shaped graph, and understanding its behavior helps in plotting parts of the piecewise function involving absolute values.

Graphing Linear Functions

Linear functions have the form f(x) = mx + b and graph as straight lines. For the piecewise function, each piece is linear or absolute value-based, so knowing how to plot lines and interpret slopes and intercepts is crucial for accurate graphing.