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Graphing Piecewise Non-Linear Functions
This lesson shows how to graph piecewise functions whose pieces are curves, such as a piecewise quadratic function, rather than lines. Split the domain, check whether boundary points are open or closed, and sketch each curved piece within its own restricted domain, with two fully worked examples.
Steps for Graphing a Piecewise Non-Linear Function
- Write down the domain restriction attached to each piece, the same way you would when you write domain and range for a single function.
- Identify what kind of curve each piece is: a parabola, a square root, an exponential curve, or something else.
- Substitute the boundary x-value into each piece that includes it to find the exact point where the pieces meet.
- Decide whether each boundary point is open (not included, drawn as a hollow circle) or closed (included, drawn as a solid circle).
- Sketch each curve as if it covered the whole number line, then erase or ignore the portion outside its assigned domain.
- Draw the remaining pieces together on the same axes to form the full graph.
Example 1: A Continuous Piecewise Quadratic Function
Graph \( f(x) = \begin{cases} x^2 & \)if \( x \le 1 \)
\( \sqrt{x} & \)if \( x > 1 \end{cases} \).
Both pieces meet at \(x = 1\). Substituting gives \(f(1) = 1^2 = 1\) from the first rule, since \(x \le 1\) includes 1, so this point is closed. The second rule only starts just after \(x = 1\), so its point at \((1, 1)\) is technically open, but since it lands on the exact same spot as the closed point from the first piece, the graph looks perfectly connected there.
Graph the parabola \(x^2\), the same way you would in graphing parabolas for given quadratic functions, but keep only the part with \(x \le 1\):
Then graph the square root piece, keeping only the part with \(x > 1\):
Placing both pieces on the same axes gives the full graph of \(f(x)\): a rising parabola arc up to \((1, 1)\), which then flows into a slower-rising square root curve.
Example 2: A Piecewise Function with a Jump Discontinuity
Graph \( g(x) = \begin{cases} -x^2 + 4 & \)if \( x < 0 \)
\( 2^x & \)if \( x \ge 0 \end{cases} \).
Check the boundary at \(x = 0\) in both pieces. The first piece gives \(-0^2 + 4 = 4\), but since its domain is \(x < 0\), the point \((0, 4)\) is not actually reached, so it gets an open circle. The second piece gives \(2^0 = 1\), and since its domain is \(x \ge 0\), the point \((0, 1)\) is included, so it gets a closed circle. Because \(4 \ne 1\), the graph jumps at \(x = 0\) instead of connecting smoothly.
The second piece is an exponential curve. Graph it the same way you would for any exponential function, using the same techniques as in graphing exponential functions, then keep only \(x \ge 0\):
Combining both sketches shows a downward-curving parabola ending in an open circle at \((0, 4)\), with a separate exponential curve starting at a closed circle at \((0, 1)\) and rising slowly to the right. This visible gap is called a jump discontinuity.
Common Mistakes to Avoid
- Do not draw a whole curve first and trim it later without checking the domain restriction carefully; it is easy to keep the wrong side of the boundary.
- Do not assume the pieces always connect. Always plug the boundary value into both pieces separately to see if the graph is continuous or has a jump.
- Do not forget which piece owns the boundary point. Only one rule can be evaluated at that exact x-value, and that rule decides whether the circle is open or closed.
- Do not mix up the shapes of different curve families. A square root curve, a parabola, and an exponential curve each have distinctly different growth patterns.