Slope of Secant Line: Formula & Examples

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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.

Table of Content

What is a Secant Line?
Slope of Secant Line
Slope of Secant Line Formula
Things to Remember
Sample Questions

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.

Secant 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".

Also Read:

Trigonometry Values: Identities, Ratios and Formula

Circles: Definition, Properties, Theorems and Examples

Slope of Secant Line Formula

Need: Slope Formula between the two points (c, f(c)) and (c + ?x, f(c + ?x)).

Slope of Secant Line Formula

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.

Also Read:

Tangent to a Circle: Definition, Equations, Theorems and Examples	Areas Related To Circles Formula
Determinant of a MatrixDeterminant of a Matrix	Euclid's Division LemmaEuclid's Division Lemma
L' Hospital RuleL' Hospital Rule	Rationalise the DenominatorRationalise the Denominator
Class MarkTangent	Probability Impossible EventsCones
Circle: Definition, Equation and Segments	Some Applications of Trigonometry MCQs

Slope of Secant Line: Formula & Examples

What is a Secant Line?

Slope of Secant Line

Slope of Secant Line Formula

Points to Remember

Sample Questions

CBSE CLASS XII Related Questions

2.
Check whether \[ f:\mathbb{R}-\{3\}\rightarrow\mathbb{R} \] defined as \[ f(x)=\frac{x-2}{x-3} \] is onto or not.

3.
The domain of \[ f(x)=\cos^{-1}(2x-5) \] is:

4.
For a square matrix \(A\), \[ (3A)^{-1}= \]

5.
If \[ B(\operatorname{adj} B)= \begin{bmatrix} \frac{1}{3} & 0 & 0\\ 0 & \frac{1}{3} & 0\\ 0 & 0 & \frac{1}{3} \end{bmatrix}, \] then the value of \[ \det(B^{-1}) \] is:

6.
A function \[ f:\mathbb{R}-\left\{\frac{3}{5}\right\} \to \mathbb{R}-\left\{\frac{3}{5}\right\} \] is defined as \[ f(x)=\frac{3x+2}{5x-3}. \] Show that \(f\) is one-one and onto.

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Slope of Secant Line: Formula & Examples

What is a Secant Line?

Slope of Secant Line

Slope of Secant Line Formula

Points to Remember

Sample Questions

CBSE CLASS XII Related Questions

2.Check whether \[ f:\mathbb{R}-\{3\}\rightarrow\mathbb{R} \] defined as \[ f(x)=\frac{x-2}{x-3} \] is onto or not.

3.The domain of \[ f(x)=\cos^{-1}(2x-5) \] is:

4. For a square matrix \(A\), \[ (3A)^{-1}= \]

5.If \[ B(\operatorname{adj} B)= \begin{bmatrix} \frac{1}{3} & 0 & 0\\ 0 & \frac{1}{3} & 0\\ 0 & 0 & \frac{1}{3} \end{bmatrix}, \] then the value of \[ \det(B^{-1}) \] is:

6. A function \[ f:\mathbb{R}-\left\{\frac{3}{5}\right\} \to \mathbb{R}-\left\{\frac{3}{5}\right\} \] is defined as \[ f(x)=\frac{3x+2}{5x-3}. \] Show that \(f\) is one-one and onto.

Comments

SUBSCRIBE TO OUR NEWS LETTER

2.
Check whether \[ f:\mathbb{R}-\{3\}\rightarrow\mathbb{R} \] defined as \[ f(x)=\frac{x-2}{x-3} \] is onto or not.

3.
The domain of \[ f(x)=\cos^{-1}(2x-5) \] is:

4.
For a square matrix \(A\), \[ (3A)^{-1}= \]

5.
If \[ B(\operatorname{adj} B)= \begin{bmatrix} \frac{1}{3} & 0 & 0\\ 0 & \frac{1}{3} & 0\\ 0 & 0 & \frac{1}{3} \end{bmatrix}, \] then the value of \[ \det(B^{-1}) \] is:

6.
A function \[ f:\mathbb{R}-\left\{\frac{3}{5}\right\} \to \mathbb{R}-\left\{\frac{3}{5}\right\} \] is defined as \[ f(x)=\frac{3x+2}{5x-3}. \] Show that \(f\) is one-one and onto.