Use the inverse of the function y = x2 − 18x to find the unknown values.
b=
c=
d=

To find the inverse of the function \( y = x^2 - 18x \), we will first complete the square to express the function in a form that allows us to find its inverse. Here’s the step-by-step process:

### Step 1: Express the function and complete the square
The given function is:

\[
y = x^2 - 18x
\]

We need to complete the square for the quadratic term. To do that:

1. Take half of the coefficient of \(x\), which is \(-18\), and divide by 2:
   \[
   \frac{-18}{2} = -9
   \]

2. Square this value:
   \[
   (-9)^2 = 81
   \]

3. Add and subtract 81 inside the expression to complete the square:
   \[
   y = (x^2 - 18x + 81) - 81 = (x - 9)^2 - 81
   \]

Thus, the function becomes:
\[
y = (x - 9)^2 - 81
\]

### Step 2: Solve for the inverse
Now, swap \(x\) and \(y\) to solve for the inverse function:

\[
x = (y - 9)^2 - 81
\]

Add 81 to both sides:

\[
x + 81 = (y - 9)^2
\]

Take the square root of both sides:

\[
\sqrt{x + 81} = y - 9
\]

Finally, solve for \(y\):

\[
y = \sqrt{x + 81} + 9
\]

### Step 3: Interpretation and solving for b, c, d
To find the unknown values \(b\), \(c\), and \(d\), the problem should have provided values for \(x\) that correspond to \(y = b\), \(y = c\), and \(y = d\). Can you clarify or provide the values of \(x\) that relate to each? This way, we can substitute them into the inverse function to find \(b\), \(c\), and \(d\).

---

Do you have any questions or would you like further details on any part of the process? 

Here are 5 related questions for deeper understanding:
1. How can we verify that this inverse function is correct?
2. How do we determine the domain and range of the inverse function?
3. What is the significance of completing the square in this process?
4. How does the inverse of a quadratic differ from linear functions?
5. Can we always find an inverse for quadratic functions?

**Tip:** When solving for inverses of functions, it’s important to ensure that the function is one-to-one over the interval you're considering.

Use the inverse of the function y = x^2 − 18x to find the unknown values.
b=
c=
d=

Learn how to find the inverse of the quadratic function y = x^2 − 18x by completing the square and solving for unknown values. This tutorial covers key algebraic concepts, such as completing the square and inverse functions, suitable for students in Grades 9-12.

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