Question

How do you find the slope of the secant lines of $y = \sqrt x$ through the points: $x = 1$ and $x = 4$?

Answer 1

The slope of a line that passes through the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is represented by $m$ and calculated as $m = \dfrac{{\left( {{y_2} - {y_1}} \right)}}{{\left( {{x_2} - {x_1}} \right)}}$. Also the secant of a curve is a line that cuts the curve at least two points.

Complete step by step solution:
The given equation is $y = \sqrt x$.

We have to find the slope of a secant for the given equation $y = \sqrt x$. Secant is a line that passes through at least two points of a curve.

It is given in the question that required secant line cuts the curve $y = \sqrt x$ at points $x = 1$ and $x = 4$.

Let’s find the point at which $x = 1$ cuts the equation $y = \sqrt x$ as shown below.

Substitute $1$ as $x$ in the equation $y = \sqrt x$ and solve for $y$ as follows:
$\Rightarrow y = \sqrt 1$
$\Rightarrow y = 1$

So, one of the points is $\left( {1,1} \right)$ on the curve from where the secant line passes.

Similarly, obtain the point at which $x = 4$ cuts the equation $y = \sqrt x$ as shown below.

Substitute $4$ as $x$ in the equation $y = \sqrt x$ and solve for $y$ as follows:
$\Rightarrow y = \sqrt 4$
$\Rightarrow y = 2$

So, the other point is $\left( {4,2} \right)$ on the curve from where the secant line passes.

Therefore, the secant line of a given curve passes through points $\left( {4,2} \right)$ and $\left( {1,1} \right)$.

Now, obtain the slope of the secant line by the use of slope of a line formula as shown below.

Substitute point $\left( {{x_1},{y_1}} \right)$ as $\left( {4,2} \right)$ and point $\left( {{x_2},{y_2}} \right)$ as $\left( {1,1} \right)$ in the slope formula $m = \dfrac{{\left( {{y_2} - {y_1}} \right)}}{{\left( {{x_2} - {x_1}} \right)}}$ and obtain the value of a slope as shown below.

$\Rightarrow m = \dfrac{{\left( {1 - 2} \right)}}{{\left( {1 - 4} \right)}}$
$\Rightarrow m = \dfrac{{ - 1}}{{ - 3}}$
$\Rightarrow m = \dfrac{1}{3}$

Thus, the slope of a secant line that passes through the points $x = 1$ and $x = 4$ of the equation $y = \sqrt x$ is $\dfrac{1}{3}$.

Note: The slope formula of a line is also called the tangent to the line. For any curve, the tangent to the curve is derivative of the curve and it varies point to point along a curve whereas tangent of line is a constant and it does not vary with point to point along a line.