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Secant line is always a straight line which connects two different locations on a function. It can also be referred to as the average rate of change or the slope between two points. The slope between two points and the average rate of change of a function between two points is the same thing. A line's slope is defined as the ratio of rise over run. Secant line in any curve is that line which is used to connect any two points in that curve. When one of these locations approaches the other, the slope of the secant line changes to the slope of the tangent line at that moment.
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Key Takeaways: Ratio, Secant, Slope, Angle, Tangent, Chord, Diameter, Radius
What is a Secant Line?
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Secant line, which is also known as a secant, is a straight line which is used to connect two points on a curve. The secant line tends to a tangent line when the two points are brought together (or, more correctly, as one is brought towards the other). Consider the graph of y = x² with a secant line shown below, where x denotes the graph's horizontal line and y denotes the graph's vertical line.
Slope of Secant Line
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Because a secant line is also a line, you may apply the formula to calculate its slope. Depending on the available data, there are several formulas for calculating the slope of a secant line.
Let’s take a curve namely y which is equal to f(x). Also take a secant line drawn to this curve. If there are (X1, Y1) and (X2, Y2) are two points on the curve then y=f(x) Through which the secant line passing through then:
| The slope of the secant line = Y2 - Y1/X2 - X1 |
If the secant line is passing through two points (a, f(a)) and (b, f(b)), then:
| Slope of the secant line = f(b) − f(a)/b-a |
This is also referred to as "average rate of change of f(x)" from x = a to x = b.
If the secant line is passing through two points P and Q,where P = (x, f(x)) and Q = (x + h, f(x + h)), then:
| Slope of the secant line = f(x+h) - f(x)/h |
Here, f(x+h) - f(x)/h is also referred to as the "difference quotient".
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Slope of Secant Line Formula
Need: Slope Formula between the two points (c, f(c)) and (c + ?x, f(c + ?x)).
Points to Remember
Following are some important points:
- The tangent line formula is equal to the slope of the secant line formula (which is also nothing but the derivative of the function at that point).
- The average rate of change of (f) across the interval is also known as the slope of the secant line.
- Some common examples of secant lines are Chords, Diameter, Radius, etc.
- Secant is derived from the Latin word secare, which means "to cut." A secant intersects a circle at exactly two locations in the case of a circle.
- Slope of f(x) at x =1 of these secant lines is the limit of the slopes.
Sample Questions
Ques: Find the slope of the secant line of a function that passes through the points (3, 10) and (-2, 19). (3 Marks)
Ans: The given points on the secant line are:
(X1, Y1) = (3,10)
(X2, Y2) = (-2, 19)
Using the slope of the secant line formula,
The slope of the secant line
=(Y2 - Y1)/(X2 - X1)
= (19 - 10) / (-2 - 3)
= 9 / (-5) (or) - 9/5
The slope of the secant line = -9/5.
Ques: Find the slope of the secant line of the function f(x) = x² - 3 that passes through the points (2, f(2)) and (3, f(3)) using the slope of the secant line formula.(3 Marks)
Ans: f(2) = 2² - 3 = 1.
f(3) = 3² - 3 = 6.
The slope of the secant line
= f(3) - f(2) / 3 - 2
= (6 - 1) / (1)
= 5
The slope of the secant line = 5.
Ques: Find the slope of the secant line of a function that passes through the points (7, 12) and (-5, 10). (3 Marks)
Ans: The given points on the secant line are:
(X1, Y1) = (7, 12)
(X2, Y2) = (-5, 10)
Using the slope of the secant line formula,
The slope of the secant line
=(Y2 - Y1)/(X2 - X1)
= (10 - 12) / (-5 - 7)
= -2 / (-12) (or) - 1/6
The slope of the secant line = -1/6.
Ques: Find the slope of the secant line of a function that passes through the points (8, 12) and (9, 10). (3 Marks)
Ans: The given points on the secant line are:
(X1, Y1) = (8, 12)
(X2, Y2) = (9, 10)
Using the slope of the secant line formula,
The slope of the secant line
=(Y2 - Y1)/(X2 - X1)
= (10 - 12) / (9 - 8)
= -2 / 1 (or) - 2/1
The slope of the secant line = -2/1.
Ques: Find the slope of the secant line of the function f(x) = x² - 3 that passes through the points (2, f(5)) and (3, f(2)) using the slope of the secant line formula. (3 Marks)
Ans: f(5) = 5² - 3 = 22.
f(2) = 2² - 3 = 1.
The slope of the secant line
= f(2) - f(5) / 5 - 2
= (1 - 22) / (1)
= - 21
The slope of the secant line = -21
Ques: Find the slope of the secant line of a function that passes through the points (5, 7) and (9, 3). (3 Marks)
Ans: The given points on the secant line are:
(X1, Y1) = (5, 7)
(X2, Y2) = (9, 3)
Using the slope of the secant line formula,
The slope of the secant line
=(Y2 - Y1)/(X2 - X1)
= (3 - 7) / (9 - 5)
= -4 / 4 (or) - 1/1
The slope of the secant line = -1/1.
Ques: Find the slope of the secant line of the function f(x) = x³ - 3 that passes through the points (2, f(4)) and (10, f(3)) using the slope of the secant line formula.(3 Marks)
Ans: f(4) = 4³ - 10 = 54.
f(3) = 3³ - 2 = 25.
The slope of the secant line
= f(3) - f(4) / 4 - 3
= (25 - 54) / (1)
= - 29
The slope of the secant line = -29
Ques: Find the slope of the secant line of a function that passes through the points (4, 6) and (7, 10). (3 Marks)
Ans: The given points on the secant line are:
(X1, Y1) = (4, 6)
(X2, Y2) = (7, 10)
Using the slope of the secant line formula,
The slope of the secant line
=(Y2 - Y1)/(X2 - X1)
= (10 - 6) / (7 - 4)
= 4 / 3 (or) 4/3
The slope of the secant line = 4/3.
Ques: Find the slope of the secant line of a function that passes through the points (5, 7) and (9, 10). (3 Marks)
Ans: The given points on the secant line are:
(X1, Y1) = (5, 7)
(X2, Y2) = (9, 10)
Using the slope of the secant line formula,
The slope of the secant line
=(Y2 - Y1)/(X2 - X1)
= (10 - 7) / (9 - 5)
= 3 / 4 (or) 3/4
The slope of the secant line = 4/3.
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