To find the equation of the tangent line to the graph of a function [tex]\(g\)[/tex] at a particular point [tex]\(x = a\)[/tex], we use the formula for the equation of a line in point-slope form:
[tex]\[ y - g(a) = g'(a) (x - a) \][/tex]
Here, we are given the following information:
- [tex]\(g(7) = -3\)[/tex], which means [tex]\(g(a) = -3\)[/tex] when [tex]\(a = 7\)[/tex].
- [tex]\(g'(7) = -1\)[/tex], which means the slope of the tangent line [tex]\(g'(a) = -1\)[/tex] when [tex]\(a = 7\)[/tex].
Using the point-slope form equation:
[tex]\[ y - g(7) = g'(7) (x - 7) \][/tex]
Substitute the given values into the equation:
[tex]\[ y - (-3) = -1 (x - 7) \][/tex]
This simplifies to:
[tex]\[ y + 3 = -1 (x - 7) \][/tex]
Distribute the slope on the right-hand side:
[tex]\[ y + 3 = -x + 7 \][/tex]
Isolate [tex]\(y\)[/tex] to find the equation in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y = -x + 7 - 3 \][/tex]
[tex]\[ y = -x + 4 \][/tex]
Therefore, the equation of the tangent line to the graph of [tex]\(g\)[/tex] at [tex]\(x = 7\)[/tex] is:
[tex]\[ y = -x + 4 \][/tex]
In this form, the y-intercept is 4 and the slope of the line is -1. The complete equation thus characterizes the tangent line at the specified point.
