Let's find the constants [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] given the function [tex]\( y = x^2 - 18x \)[/tex].
1. Rewrite the quadratic equation:
The initial function is given by:
[tex]\[
y = x^2 - 18x
\][/tex]
We want to find the inverse function. Start by rewriting the function in a quadratic equation form:
[tex]\[
x^2 - 18x + y = 0
\][/tex]
2. Solve for [tex]\(x\)[/tex] using the quadratic formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation, [tex]\( a = 1 \)[/tex], [tex]\( b = -18 \)[/tex], and [tex]\( c = y \)[/tex]. Plugging these values into the quadratic formula, we get:
[tex]\[
x = \frac{18 \pm \sqrt{18^2 - 4 \cdot 1 \cdot y}}{2 \cdot 1}
\][/tex]
Simplifying inside the square root:
[tex]\[
x = \frac{18 \pm \sqrt{324 - 4y}}{2}
\][/tex]
Simplifying further:
[tex]\[
x = 9 \pm \frac{\sqrt{324 - 4y}}{2}
\][/tex]
3. Express [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex]:
In terms of inverse, we write this in the form:
[tex]\[
y = \pm \sqrt{bx + c} + d
\][/tex]
4. Determine the constants [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex]:
By comparing:
[tex]\[
x = 9 \pm \frac{\sqrt{324 - 4y}}{2}
\][/tex]
to the general form:
[tex]\[
y = \pm \sqrt{bx + c} + d
\][/tex]
We find the constants by matching terms:
- The coefficient of [tex]\( y \)[/tex] inside the square root is [tex]\( -4 \)[/tex], so [tex]\( b = -4 \)[/tex].
- The constant term inside the square root is [tex]\(324\)[/tex], so [tex]\( c = 324 \)[/tex].
- The term added outside the square root is [tex]\( 9 \)[/tex], so [tex]\( d = 9 \)[/tex].
Thus, the constants [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] are:
[tex]\[
\begin{array}{l}
b = -4 \\
c = 324 \\
d = 9
\end{array}
\][/tex]
