Graph Radical Functions - Algebra-2

1. Fundamental Concepts

Definition: Radical functions are functions containing radicals (e.g., square root \(y = \sqrt{x}\), cube root \(y = \sqrt[3]{x}\)), where the independent variable lies inside the radical.
Domain and Range:
- Square root function: Domain is \(x \geq 0\) (i.e., \([0, +\infty)\)), and range is \(y \geq 0\) (i.e., \([0, +\infty)\));
- Cube root function: Both domain and range are all real numbers (i.e., \((-\infty, +\infty)\)).
Key Coordinates of the Parent Function (taking \(y = \sqrt{x}\) as an example): \((0, 0)\), \((1, 1)\), \((4, 2)\), \((9, 3)\). The graph starts at \((0, 0)\) and rises gently to the right.

Graph Features:
- Square root function: Exists only in the first quadrant, starts at \((0, 0)\), increases monotonically with a decreasing growth rate;
- Cube root function: Passes through the origin, has an "S" shape, spans the third and first quadrants, and is symmetric about the origin.
Monotonicity: The square root function is monotonically increasing on \([0, +\infty)\); the cube root function is monotonically increasing on \(\mathbb{R}\) (all real numbers).
Special Points: For the square root function, \((0, 0)\) is the starting point of both the domain and range; for the cube root function, \((0, 0)\) is the center of symmetry and the intersection point with the coordinate axes.

Determine the domain: \(x \geq 0\);
Plot the key points \((0, 0)\), \((1, 1)\), and \((4, 2)\) on the coordinate plane;
Connect these points with a smooth curve, which rises to the right in the first quadrant.

Find the domain: For the square root to be defined, \(x - 2 \geq 0\), so \(x \geq 2\);
Calculate and plot key points: When \(x = 2\), \(y = 0\) (point \((2, 0)\)); when \(x = 3\), \(y = 1\) (point \((3, 1)\)); when \(x = 6\), \(y = 2\) (point \((6, 2)\));
Connect the plotted points with a smooth curve. The shape of the graph is the same as that of \(y = \sqrt{x}\), with the starting point at \((2, 0)\).

Find the domain: The radicand \(-x + 4\) must be non-negative for the square root to be real, so \(-x + 4 \geq 0\), which simplifies to \(x \leq 4\);
Find the range: Let \(t = \sqrt{-x + 4}\). Since the square root of a real number is non-negative, \(t \geq 0\). Then \(y = t + 1\), so \(y \geq 0 + 1 = 1\), meaning the range is \(y \geq 1\) (i.e., \([1, +\infty)\));
Check if the point \((-5, 4)\) is on the graph: Substitute \(x = -5\) into the function. Calculate \(y = \sqrt{-(-5) + 4} + 1 = \sqrt{5 + 4} + 1 = \sqrt{9} + 1 = 3 + 1 = 4\). Since the calculated y-value equals the y-coordinate of the point, \((-5, 4)\) lies on the graph.

Determine the Domain: For even-index radicals (e.g., square roots), the radicand must be non-negative (\(\geq 0\)); for odd-index radicals (e.g., cube roots), there is no such constraint. Solve the corresponding inequality to obtain the domain.
Graph the Function: First, determine the domain; second, select easy-to-calculate values of x within the domain, compute the corresponding y-values; third, plot the resulting points on the coordinate plane; finally, connect the points with a smooth curve.
Verify if a Point Lies on the Graph: First, check if the x-coordinate of the point is within the domain of the function. If not, the point is not on the graph. If it is, substitute the x-coordinate into the function, calculate the corresponding y-value, and compare it with the y-coordinate of the point. If they are equal, the point is on the graph; otherwise, it is not.
Find the Range: Use substitution (e.g., let t be the radicand), determine the range of t based on the type of radical, and then derive the range of y from the range of t.