Slope of the Secant Line Formula
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 = (y2 β y1)/(x2 β x1)
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) β f(a))/(b β 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 β y1)/(x2 β x1)
= (5 β 11)/(2 β 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 β y1)/(x2 β x1)
=> 7 = (6 β 3)/(1 β x)
=> 7 = 3/(1 β x)
=> 7 β 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 β y1)/(x2 β x1)
=> 4 = (y β 4)/(3 β 5)
=> 4 = (y β 4)/(-2)
=> -8 = y β 4
=> y = -4
Problem 4. Calculate the slope of a secant line for the function f(x) = x2 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) β f(a))/(b β a)
= (f(5) β f(3))/ (5 β 3)
= (25 β 9)/2
= 16/2
= 8
Problem 5. Calculate the slope of a secant line for the function f(x) = 4 β 3x3 that joins the two points (1, f(1)) and (2, f(2)).
Solution:
We have, f(x) = 4 β 3x3
Calculate the value of f(1) and f(2).
f(3) = 4 β 3(1)3 = 4 β 3 = 1
f(5) = 4 β 3(2)3 = 4 β 24 = -20
Using the formula, we have
m = (f(b) β f(a))/(b β a)
= (f(2) β f(1))/ (2 β 1)
= -20 β 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, (x1, y1) = (x, 7), (x2, y2) = (9, 2) and m = 5.
Using the formula, we have
m = (y2 β y1)/(x2 β x1)
=> 5 = (2 β 7)/(9 β x)
=> 5 = -5/(9 β x)
=> 45 β 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, (x1, y1) = (1, 5), (x2, y2) = (8, y) and m = 9
Using the formula, we have
m = (y2 β y1)/(x2 β x1)
=> 9 = (y β 5)/(8 β 1)
=> 9 = (y β 5)/7
=> y β 5 = 63
=> y = 68