Question

How do you find the equation of the secant line of $f\left( x \right) = {x^2} - 5x$ through the points $\left[ {1,8} \right]$ ?

Answer 1

Here in this question, we have to find the equation of the secant line passing through the two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$. First find the ${y_1}$ and ${y_2}$ on substituting ${x_1} = 1$ and ${x_2} = 8$ in the given equation. Then find the equation by using the Point-Slope formula $y - {y_1} = m\left( {x - {x_1}} \right)$ before finding the equation first we have to find the slope using the formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$. On simplification to the point-slope formula we get the required solution.

Complete step by step solution:
A secant line is simply a linear equation and with two given points we can find the equation.Consider, the given equation of secant line
$f\left( x \right) = {x^2} - 5x$-------(1)
Now, find the two points on the secant line
${x_1} = 1$ then
${y_1} = {1^2} - 5(1)$
$\Rightarrow \,\,{y_1} = 1 - 5$
$\Rightarrow \,\,{y_1} = - 4$
The point $\left( {{x_1},{y_1}} \right) = \left( {1, - 4} \right)$ and
${x_2} = 8$ then
${y_1} = {8^2} - 5(8)$
$\Rightarrow \,\,{y_1} = 64 - 40$
$\Rightarrow \,\,{y_1} = 24$
The point $\left( {{x_2},{y_2}} \right) = \left( {8,24} \right)$ .
Now, we have to find the equation of secant line passing through the points $\left( {1, - 4} \right)$ and $\left( {8,24} \right)$ by using the slope-point formula $y - {y_1} = m\left( {x - {x_1}} \right)$-------(2)
Before this, find the slope $m$in point-slope formula by using the formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Where ${x_1} = 1$, ${x_2} = 8$, ${y_1} = - 4$ and ${y_2} = 24$ on substituting this in formula, then
$m = \dfrac{{24 - \left( { - 4} \right)}}{{8 - 1}}$
$\Rightarrow \,\,\,m = \dfrac{{24 + 4}}{{8 - 1}}$
$\Rightarrow \,\,\,m = \dfrac{{28}}{7}$
On simplification, we get
$m = 4$
Now we get the gradient or slope of the line which passes through the points $\left( {1, - 4} \right)$ and $\left( {8,24} \right)$. Substitute the slope m and the point $\left( {{x_1},{y_1}} \right) = \left( {1, - 4} \right)$ in the point slope formula. Consider the equation (2)
$y - {y_1} = m\left( {x - {x_1}} \right)$
Where $m = 4$, ${x_1} = 1$ and ${y_1} = - 4$ on substitution, we get
$y - \left( { - 4} \right) = 4\left( {x - 1} \right)$
$\Rightarrow \,\,y + 4 = 4x - 4$
Subtract 4 on both side, then
$y + 4 - 4 = 4x - 4 - 4$
On simplification, we get
$\therefore\,\,y = 4x - 8$

Hence, the equation of the secant line passing through points $\left( {1, - 4} \right)$ and $\left( {8,24} \right)$ is $y = 4x - 8$.

Note: To determine the equation of secant line, we consider the equation of line and the slope of the line is determined. Here we use simple arithmetic operations while simplifying the equation and we should take care of the sign of the terms of the equation.A secant of a curve is a line that intersects the curve at a minimum of two distinct points.