Slope of the Secant Line Formula, Definition, Solved Examples

Slope of the Secant Line Formula: Prior to discussing the formula for the slope of the secant line, it's essential to revisit the definitions of slope and a secant. The slope of a line is represented as the ratio of its rise over run. In the field of curves, a secant line is a straight line that intersects two points on the curve. As these points draw closer, the slope of the secant line approaches the slope of the tangent line at that specific point.

Slope of the Secant Line Formula

The formula for the slope of a secant line is derived from the formula for the slope of a line because a secant line is, essentially, a line. There exist various formulas to determine the slope of a secant line based on the available information. Consider a curve represented by y = f(x) and envision a secant line drawn across this curve.

If two points on the curve, denoted as (x ₁ , y ₁ ) and (x ₂ , y ₂ ), define the path of the secant line intersecting the curve y = f(x), then the slope of the secant line can be calculated using the formula:

Slope of the secant line = (y ₂ - y ₁ ) / (x ₂ - x ₁ )

In cases where the secant line passes through two specific points, namely (a, f(a)) and (b, f(b)), the slope of the secant line can be determined using:

Slope of the secant line = (f(b) - f(a)) / (b - a)

This is also known as the "average rate of change of f(x)" from x = a to x = b.

Moreover, if the secant line traverses through two points, P and Q, where P = (x, f(x)) and Q = (x + h, f(x + h)), then the calculation involves:

Important point to note: The formula for the slope of the secant line provides the formula for the slope of the tangent line (which is essentially the derivative of the function at that specific point) when

x ₂ approaches x ₁ , b approaches a, or h approaches 0.

Now, let's explore a few solved examples to gain a better understanding of the secant line formula.

Slope of the Secant Line Formula Solved Examples

Example 1: Determining the slope of the secant line for the function f(x)=4x−5 passing through the points(1, f(1)) and (4, f(4))  using the slope of the secant line formula.

Solution: Given the function f(x)=4x−5:

Substituting x = 1 into f(x) gives

f(1)=4(1)−5=−1.

Substituting x = 4 into f(x) gives

f(4)=4(4)−5=11.

Now, applying the slope of the secant line formula:

Slope= f(4)−f(1) ​/ 4−1

= 11−(−1) ​/ 3

= 12 ​/ 3

=4

Answer: The slope of the secant line = 4.

Example 2: Determining the slope of the secant line for the function g(x)=x ² +2x  passing through the points (2, g(2)) and  (5, g(5))   using the slope of the secant line formula.

Solution: Given the function g(x)=x ² +2x:

Substituting x = 2 into g(x) gives g(2)=2 ² +2(2)=8.

Substituting x = 5 into g(x) gives g(5)=5 ² +2(5)=35.

Applying the slope of the secant line formula:

Slope= g(5)−g(2) / 5−2 ​ = 35−8 / 3 ​ = 27/3 ​ =9

Answer: The slope of the secant line = 9.

Example 3: h(x)=3x−7 Points: (0, h(0)) and (2, h(2))

Solution: Given the function h(x)=3x−7:

Substituting x = 0 into h(x) gives h(0)=3(0)−7=−7.

Substituting x = 2 into h(x) gives h(2)=3(2)−7=−1.

Using the slope of the secant line formula:

Slope= h(2)−h(0) / 2−0 ​

= −1−(−7) ​/ 2

= 6 ​/ 2

=3

Answer: The slope of the secant line = 3.

Example 4: Function: k(x)=2x ² −3x Points: (1, k(1)) and (3, k(3))

Solution: Given the function k(x)=2x ² −3x:

Substituting x = 1 into k(x) gives

k(1)=2(1) ² −3(1)= -1.

Substituting x = 3 into k(x) gives

k(3)=2(3) ² −3(3)=15.

Applying the slope of the secant line formula:

Slope= k(3)−k(1) ​ / 3−1

= 15−(−1) ​ / 2

= 16 / 2 ​

=8

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