Question

graph \\(y = \sqrt{x}\\)
graph \\(y = \sqrt4{x}\\)
graph \\(y = \sqrt6{x}\\)

Explanation:

🆕 New Concept Discovered: Graphing Radical Functions
How even-index radical curves behave and compare.

Step 1: Identify key characteristics and domain

For any radical function with an even index, \(y = \sqrt[n]{x}\) (where \(n = 2, 4, 6\)):

• Domain: Since we cannot take an even root of a negative number in the real number system, we must have \(x \ge 0\).
• Range: The principal root is always non-negative, so \(y \ge 0\).
• Key Points: All three graphs pass through the points \((0, 0)\) and \((1, 1)\) because:

$$ \sqrt[n]{0} = 0 \quad \text{and} \quad \sqrt[n]{1} = 1 $$

Step 2: Compare the growth rates

To see how the curves behave relative to each other, we look at two intervals:

• Between \(0\) and \(1\) (\(0 < x < 1\)):

Higher roots of a fraction make it larger. For example, at \(x = 0.01\):

$$ \sqrt{0.01} = 0.1 \quad \text{and} \quad \sqrt[4]{0.01} \approx 0.316 $$

Thus, for \(0 < x < 1\), the graph of \(y = \sqrt[6]{x}\) is the steepest (highest), followed by \(y = \sqrt[4]{x}\), and \(y = \sqrt{x}\) is the lowest.

• Greater than \(1\) (\(x > 1\)):

Higher roots of a number larger than \(1\) make it smaller. For example, at \(x = 64\):

$$ \sqrt{64} = 8, \quad \sqrt[4]{64} \approx 2.83, \quad \sqrt[6]{64} = 2 $$

Thus, for \(x > 1\), the graph of \(y = \sqrt{x}\) is the highest, followed by \(y = \sqrt[4]{x}\), and \(y = \sqrt[6]{x}\) is the flattest (lowest).

Step 3: Plot key coordinate points

We can calculate specific points for each function to construct an accurate graph:

• For \(y = \sqrt{x}\):

$$ (0, 0), \quad (1, 1), \quad (4, 2) $$

• For \(y = \sqrt[4]{x}\):

$$ (0, 0), \quad (1, 1), \quad (16, 2) $$

• For \(y = \sqrt[6]{x}\):

$$ (0, 0), \quad (1, 1), \quad (64, 2) $$

Step 4: Sketch the curves

Using the points and comparisons above, we sketch the three curves on the same coordinate plane:

   y ^
     |
   2 +--------------------------------------------------* y = \sqrt{x}
     |                                     *
     |                       *  y = \sqrt[4]{x}
   1 +---------*------------+------------+------------+
     |       *   *
     |     *       *  y = \sqrt[6]{x}
     |   *
   0 +*----------+------------+------------+------------+--> x
     0           1            4            16           64

Answer:

To graph the three functions \(y = \sqrt{x}\), \(y = \sqrt[4]{x}\), and \(y = \sqrt[6]{x}\):

1. Start at the origin: All three curves begin at \((0,0)\) and only exist in the first quadrant (\(x \ge 0\), \(y \ge 0\)).
2. Intersection point: All three curves intersect exactly at \((1,1)\).
3. Between \(0\) and \(1\): The curve with the higher index is higher up.

$$ \sqrt[6]{x} > \sqrt[4]{x} > \sqrt{x} $$

1. For \(x > 1\): The curve with the higher index is flatter and lower down.

$$ \sqrt{x} > \sqrt[4]{x} > \sqrt[6]{x} $$