91Ó°ÊÓ

In a radical function, the endpoint is a critical point that determines where the graph begins. If you have a function in the form of \( y = a \sqrt{x - h} + k \)\, the point \( (h, k) \)\ is the endpoint.
For example, in the exercise above, the endpoint for part (a) is \( (2, 5) \), so we start by substituting \( h = 2 \)\ and \( k = 5 \)\ into the equation. This makes the function \( y = a \sqrt{x - 2} + 5 \)\.
Similarly, the endpoint for part (b) is \( (3, -2) \), leading to the function \( y = a \sqrt{x - 3} - 2 \)\.

Remember:

• The endpoint is where the function essentially 'starts' on the graph.
• Changing the endpoint shifts the graph horizontally and vertically.

To fully define the radical function, we need the value of the constant 'a'. This is done using a point that the function passes through.
In part (a) of the exercise, the function passes through \( (6, 1) \). Substituting these values into \( y = a \sqrt{x - 2} + 5 \), we get:
\[ 1 = a \sqrt{6 - 2} + 5 \]
\[ 1 = a \sqrt{4} + 5 \]
\[ 1 = 2a + 5 \]
To solve for 'a', we rearrange to find:
\[ 2a = -4 \]
\[ a = -2 \]
This gives us the final function for part (a): \( y = -2 \sqrt{x - 2} + 5 \).

Tips:

• Always substitute the point correctly into the equation.
• Simplify step by step to solve for 'a'.

For part (b) in the exercise, we tried to use an x-intercept of -6. An x-intercept is where \( y = 0 \). Substituting \( x = -6 \) into the equation for part (b):
\[ 0 = a \sqrt{-6 - 3} - 2 \]
\[ 0 = a \sqrt{-9} - 2 \]
This equation has an issue because you cannot take the square root of a negative number in the set of real numbers.
This led us to conclude that there's an inconsistency with this x-intercept in the context of radical functions.

Key Points:

• Ensure that the x-intercept provided is valid.
• The square root function only produces real output for non-negative inputs.
• If an x-intercept results in taking the square root of a negative number, further verification of the problem is needed.

Solving radical equations involves isolating the radical expression and then squaring both sides of the equation to eliminate the radical. Here are some tips:
1. Isolate the Radical: Make sure the radical expression is isolated on one side of the equation.
For example, \( \sqrt{x - 2} = 3y - 5 \).
2. Square Both Sides: Once isolated, square both sides to eliminate the radical.
\[ (\sqrt{x - 2})^2 = (3y - 5)^2 \]
3. Solve the Resulting Equation: After squaring, solve the equation that comes out of it.
4. Check for Extraneous Solutions: Squaring both sides can introduce solutions that do not satisfy the original equation. Always substitute your solutions back into the original equation to verify.

By following these steps, you can effectively tackle radical functions and their equations.
Remember to:

• Double-check your work.
• Ensure your solutions fit within the constraints of the problem.
• Practice regularly to become comfortable with these equations.