Math Problem Statement

Use the inverse of the function y = x^2 - 18x to find the unknown values for the equation y = ±√(bx + c) + d.

Solution

We are tasked with finding the inverse of the function y=x218xy = x^2 - 18x and matching it to the given form y=±bx+c+dy = \pm \sqrt{bx + c} + d.

Step 1: Completing the square

To express the given quadratic in a form that can be inverted, we start by completing the square.

The original function is: y=x218xy = x^2 - 18x

Complete the square: y=(x218x)=(x9)281y = (x^2 - 18x) = (x - 9)^2 - 81 Thus, the equation becomes: y=(x9)281y = (x - 9)^2 - 81

Step 2: Inverting the function

Now, solve for xx to find the inverse: y+81=(x9)2y + 81 = (x - 9)^2 Take the square root of both sides: ±y+81=x9\pm \sqrt{y + 81} = x - 9 Solve for xx: x=9±y+81x = 9 \pm \sqrt{y + 81}

Thus, the inverse function is: x=9±y+81x = 9 \pm \sqrt{y + 81}

Step 3: Compare with the form given

The inverse is now in the form x=±by+c+dx = \pm \sqrt{by + c} + d. Comparing it to the standard form y=±bx+c+dy = \pm \sqrt{bx + c} + d, we can match the coefficients:

  • b=1b = 1
  • c=81c = 81
  • d=9d = 9

Thus, the values are: b=1,c=81,d=9b = 1, \quad c = 81, \quad d = 9

Would you like further clarification on any step? Here are some related questions to think about:

  1. How does completing the square help in solving quadratic equations?
  2. Why do we take the square root on both sides when inverting the function?
  3. What is the general procedure for finding the inverse of a quadratic function?
  4. How do you determine the positive or negative branch of the square root when inverting?
  5. Can you find the domain and range of the inverse function?

Tip: When completing the square, always remember to balance the equation by adding and subtracting the same value to maintain equality.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Quadratic Equations
Completing the Square

Formulas

Inverse of a quadratic function: Solve y = ax^2 + bx + c for x.
Completing the square: (x - h)^2 = x^2 - 2hx + h^2

Theorems

Completing the Square Theorem
Inverse Function Theorem

Suitable Grade Level

Grades 10-12