In this lesson, we study the cube root function:
\[ f(x) = a \sqrt[3]{x - b} + c \]Use the controls below to interactively change a, b, and c. Observe vertical shifts, horizontal shifts, stretching, reflections, and intercepts.
Interactive Cube Root Graph Explorer
Domain and Range of the Cube Root Function
The cube root function is defined for all real numbers because cube roots of negative numbers exist:
\[ \sqrt[3]{-8} = -2, \quad \sqrt[3]{-27} = -3 \]This means the domain (set of possible x-values) and the range (set of possible y-values) are both:
\[ \text{Domain} = (-\infty, \infty), \quad \text{Range} = (-\infty, \infty) \]Why? Every real x has exactly one cube root, and the function grows continuously from \(-\infty\) to \(+\infty\), so the output also covers all real numbers.
Effect of Parameters
1. Vertical Shift: Parameter c
Increasing c shifts the graph upward; decreasing shifts it downward:
\[ f(x) = \sqrt[3]{x} + 2 \]2. Horizontal Shift: Parameter b
Increasing b shifts the graph to the right; decreasing shifts it left:
\[ f(x) = \sqrt[3]{x-3} \]3. Stretch / Reflection: Parameter a
- \( |a| > 1 \): vertical stretch
- \( 0 < |a| < 1 \): vertical shrink
- \( a < 0 \): reflection across x-axis
Intercepts
X-Intercept
\[ a\sqrt[3]{x-b} + c = 0 \implies x = b + \left(-\frac{c}{a}\right)^3 \]Y-Intercept
\[ f(0) = a\sqrt[3]{-b} + c \]Example: \( f(x) = 2\sqrt[3]{x-8} + 1 \)
\[ f(0) = 2\sqrt[3]{-8} + 1 = -3 \]Key Takeaways
- Cube roots exist for negative numbers.
- Domain and range are both all real numbers.
- Exactly one x-intercept.
- Parameter a controls stretch/reflection.
- Parameter b shifts horizontally.
- Parameter c shifts vertically.