Find the slope of the secant line

The calculator will find the equation of the secant line that intersects the given curve at the given points, with steps shown.

Related calculators: Line Calculator, Slope-Intercept Form Calculator with Two Points

Your Input

Find the equation of the secant line that intersects the curve $$$f{\left(x \right)} = x^{2} + 1$$$ at $$$x_{1} = 1$$$ and $$$x_{2} = 3$$$.

Solution

Find the y-coordinates of the points on the curve that correspond to the given x-coordinates.

$$$y_{1} = f{\left(x_{1} \right)} = f{\left(1 \right)} = 2$$$

$$$y_{2} = f{\left(x_{2} \right)} = f{\left(3 \right)} = 10$$$

Since we have two points, we can use the line calculator to find the equation of the secant line through the two points.

Thus, the equation of the secant line is $$$y = 4 x - 2$$$.

Answer

The equation of the secant line is $$$y = 4 x - 2$$$A.

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 – y1)/(x2 – 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) – f(a))/(b – 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 – y1)/(x2 – x1)

= (5 – 11)/(2 – 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 – y1)/(x2 – x1)

=> 7 = (6 – 3)/(1 – x)

=> 7 = 3/(1 – x)

=> 7 – 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 – y1)/(x2 – x1)

=> 4 = (y – 4)/(3 – 5)

=> 4 = (y – 4)/(-2)

=> -8 = y – 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) – f(a))/(b – a)

= (f(5) – f(3))/ (5 – 3)

= (25 – 9)/2

= 16/2

= 8

Problem 5. Calculate the slope of a secant line for the function f(x) = 4 – 3x3 that joins the two points (1, f(1)) and (2, f(2)).

Solution:

We have, f(x) = 4 – 3x3

Calculate the value of f(1) and f(2).

f(3) = 4 – 3(1)3 = 4 – 3 = 1

f(5) = 4 – 3(2)3 = 4 – 24 = -20

Using the formula, we have

m = (f(b) – f(a))/(b – a)

= (f(2) – f(1))/ (2 – 1)

= -20 – 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 – y1)/(x2 – x1)

=> 5 = (2 – 7)/(9 – x)

=> 5 = -5/(9 – x)

=> 45 – 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 – y1)/(x2 – x1)

=> 9 = (y – 5)/(8 – 1)

=> 9 = (y – 5)/7

=> y – 5 = 63

=> y = 68