To find the slope of the tangent line to the graph of the function [tex]\( g \)[/tex] at the point [tex]\((9, 2)\)[/tex], given that the tangent line passes through the point [tex]\((5, 7)\)[/tex], we follow these steps:
1. Identify Points: We have two points through which the tangent line passes:
- Point [tex]\((9, 2)\)[/tex]: This is the point on the curve where we want the slope of the tangent (denoted [tex]\( (x_1, y_1) \)[/tex]).
- Point [tex]\((5, 7)\)[/tex]: This is another point through which the tangent line passes (denoted [tex]\( (x_2, y_2) \)[/tex]).
2. Recall Slope Formula: The slope [tex]\( m \)[/tex] of the line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
3. Substitute Given Points into the Slope Formula:
- [tex]\( x_1 = 9 \)[/tex]
- [tex]\( y_1 = 2 \)[/tex]
- [tex]\( x_2 = 5 \)[/tex]
- [tex]\( y_2 = 7 \)[/tex]
Thus, the slope [tex]\( g'(9) \)[/tex] can be calculated as follows:
[tex]\[
g'(9) = \frac{7 - 2}{5 - 9}
\][/tex]
4. Perform the Calculation:
- The numerator (difference in [tex]\( y \)[/tex]-coordinates) is [tex]\( 7 - 2 = 5 \)[/tex].
- The denominator (difference in [tex]\( x \)[/tex]-coordinates) is [tex]\( 5 - 9 = -4 \)[/tex].
Therefore, the slope is:
[tex]\[
g'(9) = \frac{5}{-4} = -1.25
\][/tex]
The slope of the tangent line to the graph of function [tex]\( g \)[/tex] at point [tex]\((9, 2)\)[/tex] is
[tex]\[
g'(9) = -1.25
\][/tex]
