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[((x-a)(x-b))/((sqrt(x-c)))]=>y

[(x−a)(x−b)(x−c)]⇒y \left[\frac{(x-a)(x-b)}{(\sqrt{x-c})}\right] \Rightarrow y

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Full solution

Q. [(x−a)(x−b)(x−c)]⇒y \left[\frac{(x-a)(x-b)}{(\sqrt{x-c})}\right] \Rightarrow y
  1. Identify Transformation: Identify the transformation: We are given (x−a)(x−b)x−c\frac{(x-a)(x-b)}{\sqrt{x-c}} and we set it equal to yy, which is a standard way to represent a function.
  2. Write Standard Form: Write the function in standard form: y=(x−a)(x−b)x−cy = \frac{(x-a)(x-b)}{\sqrt{x-c}}.
  3. Check Domain Restrictions: Check for any restrictions on the domain: Since we have a square root in the denominator, x−cx-c must be greater than 00, so x>cx > c.
  4. Identify Function Behavior: Identify the behavior of the function: The function is a rational function with a square root in the denominator, which will affect its graph.
  5. Determine Vertical Asymptotes: Determine the vertical asymptotes: The vertical asymptotes occur where the denominator is zero, so we set x−c=0\sqrt{x-c} = 0, which gives us x=cx = c.
  6. Determine Zeros: Determine the zeros of the function: The zeros occur where the numerator is zero, so we set (x−a)(x−b)=0(x-a)(x-b) = 0, which gives us x=ax = a and x=bx = b.
  7. Combine Information: Combine the information: The function has vertical asymptotes at x=cx = c and zeros at x=ax = a and x=bx = b. This tells us how the function behaves around these points.

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