Sure, let's find the values of [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] in the inverse function of [tex]\( y = x^2 - 18x \)[/tex].
1. Original Function:
[tex]\[
y = x^2 - 18x
\][/tex]
2. To find the inverse function, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex], so:
[tex]\[
x = y^2 - 18y
\][/tex]
3. This is a standard quadratic equation in [tex]\( y \)[/tex]:
[tex]\[
y^2 - 18y - x = 0
\][/tex]
4. We solve this quadratic equation using the quadratic formula: [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = -18 \)[/tex], and [tex]\( c = -x \)[/tex]. Therefore:
[tex]\[
y = \frac{18 \pm \sqrt{324 + 4x}}{2}
\][/tex]
5. The inverse function, represented in the form [tex]\( y = \pm \sqrt{bx + c} + d \)[/tex], corresponds to comparing the above solution to this form. We compare:
[tex]\[
y = \frac{18 \pm \sqrt{324 + 4x}}{2} \quad \text{with} \quad y = \pm \sqrt{bx + c} + d
\][/tex]
6. Upon comparing, it fits [tex]\( y = \pm \sqrt{4x + 324} + 9 \)[/tex]. Therefore:
[tex]\[
\pm \sqrt{bx + c} + d = \pm \sqrt{4x + 324} + 9
\][/tex]
7. Thus, the values of [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] are:
[tex]\[
b = 4, \quad c = 324, \quad d = 9
\][/tex]
So the unknown values are:
[tex]\[
\begin{array}{l}
b = 4 \\
c = 324 \\
d = 9
\end{array}
\][/tex]
