Sure, here is a detailed step-by-step solution for finding a formula for the line intersecting the graph of [tex]\( f(x) = \frac{10}{x^2 + 1} \)[/tex] at [tex]\( x = 1 \)[/tex] and [tex]\( x = 3 \)[/tex].
1. Calculate [tex]\( f(1) \)[/tex] and [tex]\( f(3) \)[/tex]:
For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = \frac{10}{1^2 + 1} = \frac{10}{2} = 5
\][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = \frac{10}{3^2 + 1} = \frac{10}{9 + 1} = \frac{10}{10} = 1
\][/tex]
2. Find the coordinates of the points of intersection:
The points of intersection are:
[tex]\[
(1, f(1)) = (1, 5)
\][/tex]
[tex]\[
(3, f(3)) = (3, 1)
\][/tex]
3. Calculate the slope [tex]\( m \)[/tex] of the line passing through these points:
The slope [tex]\( m \)[/tex] is given by the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the values:
[tex]\[
m = \frac{1 - 5}{3 - 1} = \frac{-4}{2} = -2
\][/tex]
4. Use the point-slope form to find the equation of the line:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using the point [tex]\((1, 5)\)[/tex] and the slope [tex]\( m = -2 \)[/tex]:
[tex]\[
y - 5 = -2(x - 1)
\][/tex]
5. Simplify the equation to slope-intercept form ( [tex]\( y = mx + b \)[/tex] ):
[tex]\[
y - 5 = -2x + 2
\][/tex]
[tex]\[
y = -2x + 2 + 5
\][/tex]
[tex]\[
y = -2x + 7
\][/tex]
Therefore, the equation of the line intersecting the graph of [tex]\( f(x) \)[/tex] at [tex]\( x = 1 \)[/tex] and [tex]\( x = 3 \)[/tex] is:
[tex]\[
y = -2x + 7
\][/tex]
