Question

use the inverse of the function (y = x^2 - 18x) to find the unknown values.

y = pmsqrt{bx + c} + d

(b = )
(c = )
(d = )

Explanation:

<pre_analysis>
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<reasoning>

Swap variables to set up the inverse equation

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

Complete the square for the quadratic expression

\[

$$\begin{aligned} x + 81 &= y^2 - 18y + 81 \\ x + 81 &= (y - 9)^2 \end{aligned}$$

\]

Solve for y to find the inverse function

\[

$$\begin{aligned} y - 9 &= \pm\sqrt{x + 81} \\ y &= \pm\sqrt{x + 81} + 9 \end{aligned}$$

\]

Comparing this to \(y = \pm\sqrt{bx + c} + d\):
\[
b = 1,\quad c = 81,\quad d = 9
\]
</reasoning>

<answer>
Use the inverse of the function \(y = x^2 - 18x\) to find the unknown values.

\(y = \pm\sqrt{bx + c} + d\)

\(b =\) <blank>1</blank>

\(c =\) <blank>81</blank>

\(d =\) <blank>9</blank>
</answer>

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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Inverse"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Swap variables to set up the inverse equation

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

Complete the square for the quadratic expression

\[

$$\begin{aligned} x + 81 &= y^2 - 18y + 81 \\ x + 81 &= (y - 9)^2 \end{aligned}$$

\]

Solve for y to find the inverse function

\[

$$\begin{aligned} y - 9 &= \pm\sqrt{x + 81} \\ y &= \pm\sqrt{x + 81} + 9 \end{aligned}$$

\]

Comparing this to \(y = \pm\sqrt{bx + c} + d\):
\[
b = 1,\quad c = 81,\quad d = 9
\]
</reasoning>

<answer>
Use the inverse of the function \(y = x^2 - 18x\) to find the unknown values.

\(y = \pm\sqrt{bx + c} + d\)

\(b =\) <blank>1</blank>

\(c =\) <blank>81</blank>

\(d =\) <blank>9</blank>
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Inverse"
]
}
</post_analysis>