Question

How do you find the equation of the secant line through the points where \[x\] has the given values: \[f\left( x \right) = {x^2} + 2x;x = 3,x = 5\]?

Answer 1

Hint: In the given question, we have been asked to find the equation of the secant line through the points where we have been given the values of the argument. To solve this, we are going to put in the given values of the abscissa to calculate the ordinate; there are going to be two pairs. Then, we are going to find the slope of the line joining the two points. Then we are going to solve for the intercept by putting it into the standard form of the equation.

Complete step-by-step answer:
The given equation is:
\[f\left( x \right) = {x^2} + 2x\]
The given x-coordinate points are: \[x = 3,x = 5\]
Putting in the values for finding the ordinate:
\[f\left( 3 \right) = y = {3^2} + 2 \times 3 = 15\]
\[f\left( 5 \right) = y = {5^2} + 2 \times 5 = 35\]
Hence, the points are \[\left( {3,15} \right)\] and \[\left( {5,35} \right)\].
Slope of the secant line, \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{35 - 15}}{{5 - 3}} = 10\]
Now, we are going to solve for the y-intercept by putting in the known values into the standard eq:
\[y = mx + b\]
\[15 = 10 \times 3 + b\]
Thus, \[b = - 15\]
Hence, the secant equation is \[y = 10x - 15\].

Note: It is important that we know the procedure on how to apply the given information and how to use it. Without that, there is going to be no point in solving for anything. So, we must know the formulae and their applications too.