Slope of the Secant Line Formula (original) (raw)
Last Updated : 23 Jul, 2025
Understanding the Slope of the Secant Line Formula
The slope of the secant line between two points on a curve is a fundamental concept in calculus. The secant line is a straight line connecting two distinct points on the graph of a function. The slope of this line provides an average rate of change of the function between those two points.
A secant line is a straight line that connects two points on the curve of a function f(x). A secant line, also known as a secant, is basically a line that passes through two points on a curve. It tends to a tangent line when one of the two points is brought towards the other one. It is used to evaluate the equation of tangent line to a curve at a point only and only if it exists for a value (a, f(a)).
**Slope of the Secant Line Formula
The slope of a line is defined as the ratio of change in y coordinate to the change in x coordinate. If there are two points (x1, y1) and (x2, y2) connected by a secant line on a curve y = f(x) then the slope is equal to the ratio of differences between the y-coordinates to that of the x-coordinates. The slope value is represented by the symbol m.
**m = (y 2 - y 1 )/(x 2 - x 1 )
If the secant line is passing through two points (a, f(a)) and (b, f(b)) for a function f(x), then the slope is given by the formula:
**m = (f(b) - f(a))/(b - a)
**Sample Problems
**Problem 1. Calculate the slope of a secant line that joins the two points (4, 11) and (2, 5).
**Solution:
We have, (x1, y1) = (4, 11) and (x2, y2) = (2, 5)
Using the formula, we have
m = (y2 - y1)/(x2 - x1)
= (5 - 11)/(2 - 4)
= -6/(-2)
= 3
**Problem 2. The slope of a secant line that joins the two points (x, 3) and (1, 6) is 7. Find the value of x.
**Solution:
We have, (x1, y1) = (x, 3), (x2, y2) = (1, 6) and m = 7
Using the formula, we have
m = (y2 - y1)/(x2 - x1)
=> 7 = (6 - 3)/(1 - x)
=> 7 = 3/(1 - x)
=> 7 - 7x = 3
=> 7x = 4
=> x = 4/7
**Problem 3. The slope of a secant line that joins the two points (5, 4) and (3, y) is 4. Find the value of y.
**Solution:
We have, (x1, y1) = (5, 4), (x2, y2) = (3, y) and m = 4
Using the formula, we have
m = (y2 - y1)/(x2 - x1)
=> 4 = (y - 4)/(3 - 5)
=> 4 = (y - 4)/(-2)
=> -8 = y - 4
=> y = -4
**Problem 4. Calculate the slope of a secant line for the function f(x) = x 2 that joins the two points (3, f(3)) and (5, f(5)).
**Solution:
We have, f(x) = x2
Calculate the value of f(3) and f(5).
f(3) = 32 = 9
f(5) = 52 = 25
Using the formula, we have
m = (f(b) - f(a))/(b - a)
= (f(5) - f(3))/ (5 - 3)
= (25 - 9)/2
= 16/2
= 8
**Problem 5. Calculate the slope of a secant line for the function f(x) = 4 - 3x 3 that joins the two points (1, f(1)) and (2, f(2)).
**Solution:
We have, f(x) = 4 - 3x3
Calculate the value of f(1) and f(2).
f(3) = 4 - 3(1)3 = 4 - 3 = 1
f(5) = 4 - 3(2)3 = 4 - 24 = -20
Using the formula, we have
m = (f(b) - f(a))/(b - a)
= (f(2) - f(1))/ (2 - 1)
= -20 - 1
= -21
**Problem 6. The slope of a secant line that joins the two points (x, 7) and (9, 2) is 5. Find the value of x.
**Solution:
_We have, (x 1 , y 1 ) = (x, 7), (x 2 , y 2 ) = (9, 2) and m = 5.
_Using the formula, we have
_m = (y 2 - y 1 )/(x 2 - x 1 )
_=> 5 = (2 - 7)/(9 - x)
_=> 5 = -5/(9 - x)
_=> 45 - 5x = -5
_=> 5x = 50
_=> x = 10
**Problem 7. The slope of a secant line that joins the two points (1, 5) and (8, y) is 9. Find the value of y.
**Solution:
_We have, (x 1 , y 1 ) = (1, 5), (x 2 , y 2 ) = (8, y) and m = 9
_Using the formula, we have
_m = (y 2 - y 1 )/(x 2 - x 1 )
_=> 9 = (y - 5)/(8 - 1)
_=> 9 = (y - 5)/7
_=> y - 5 = 63
_=> y = 6