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

Hint: 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.