About Slope of the Secant Line Formula
Let us review what slope and secant are before learning the slope of the secant line formula. Rise over run is how a line's slope is defined. A curve's secant line is a line that connects any two points on the 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.
What do you mean by Slope of Secant Line Formula?
Because a secant line is likewise a line, we apply the slope of a line formula to determine its slope. Depending on the information given, there are several formulas for calculating the slope of a secant line. Consider the curve y = f(x) and the secant line drawn to it.
If (x1,y1)(x1,y1) and (x2,y2)(x2,y2) are two points on the curve y = f(x) through which the secant line is passing, then:
Slope of the secant line = (y2−y1)/(x2−x1)
If the secant line passes through two points (a, f(a)) and (b, f(b)), the following equation is true:
Slope of the secant line = {f(b)−f(a)}/(b−a)
From x = a to x = b, this is also known as "average rate of change of f(x)."
If the secant line passes through two points P and Q, where P = (x, f(x)) and Q = (x + h, f(x + h)), then the following is true:
Slope of secant line = {f(x+h)−f(x)}/{x+h−x} = {f(x+h)−f(x)}/h
where, {f(x+h)−f(x)}/h is also referred to as "difference quotient".
When x2→ x1, (or) b → a, (or) h 0 the slope of the secant line formula equals the slope of the tangent line formula (which is also nothing but the derivative of the function at that point).
Solved example of Slope of the Secant Line
Example: Find the slope of the secant line of a function that passes through the points (3, 10) and (-2, 19).
Sol:
It is the given points on the secant line are:
(x1,y1) = (3, 10)
(x2,y2) = (-2, 19)
Using slope of secant line formula,
Slope of secant line
= (y2−y1)/(x2−x1) = (19 - 10) / (-2 - 3)
= 9 / (-5) (or) - 9/5
Answer: slope of secant line = -9/5.
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