Question

How do you find the slope of the secant line of $ f\left( x \right) = 6.1{x^2} - 9.1x $ through the points: $ x = 8 $ and $ x = 16 $

Answer 1

Hint: In order to determine the slope of the secant line of the given function , use the standard formula for slope or average rate of change as \[slope = m = \dfrac{{\Delta y}}{{\Delta x}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]. $ {x_1},{x_2} $ will the values of $ x $ given in the question and $ {y_1},{y_2} $ are the values of function obtained by putting $ {x_1},{x_2} $ in the function respectively. Put all the values in the slope form, to get the required result.

Complete step-by-step answer:
We are given a quadratic function in variable $ x $ as $ f\left( x \right) = 6.1{x^2} - 9.1x $ .
As per the question , we have to find out the slope of the secant line to this function through the points $ x = 8 $ and $ x = 16 $ .
Before proceeding to the solution , let’s first understand what is a secant line.
So, the secant line is a straight line formed by joining two points on a function $ f\left( x \right) $ . It is simply equivalent to the average rate of change, or in other words, we can say the slope between two points on the function.
\[slope = m = \dfrac{{\Delta y}}{{\Delta x}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{f\left( {{x_2}} \right) - f\left( {{x_1}} \right)}}{{{x_2} - {x_1}}}\]
In this question we are given $ {x_1},{x_2} $ as $ 8,16 $ respectively.
Now let’s find out the value for $ f\left( {{x_1}} \right) $ by substituting all the occurrences of variable $ x $ with the value of $ {x_1} $ , we have
 $
  f\left( {{x_1}} \right) = f\left( 8 \right) \\
   \Rightarrow f\left( 8 \right) = \left( {6.1} \right){\left( 8 \right)^2} - \left( {9.1} \right)\left( 8 \right) \\
   \Rightarrow f\left( 8 \right) = 390.4 - 72.8 \\
   \Rightarrow f\left( 8 \right) = 317.6 \;
  $
Similarly find the value for $ f\left( {{x_2}} \right) $ , replacing all the $ x $ with value of $ {x_2} = 16 $ in the function
 $
  f\left( {{x_2}} \right) = f\left( {16} \right) \\
   \Rightarrow f\left( {16} \right) = \left( {6.1} \right){\left( {16} \right)^2} - \left( {9.1} \right)\left( {16} \right) \\
   \Rightarrow f\left( {16} \right) = 1561.6 - 145.6 \\
   \Rightarrow f\left( {16} \right) = 1416 \;
  $
Hence we have values as $ f\left( {{x_1}} \right) = 317.6 $ and $ f\left( {{x_2}} \right) = 1416 $ .
Now putting these values in the slope of the secant line, we get the slope as
\[
  slope = m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{f\left( {{x_2}} \right) - f\left( {{x_1}} \right)}}{{{x_2} - {x_1}}} \\
  m = \dfrac{{1416 - 317.6}}{{16 - 8}} \;
 \]
Simplifying further, we have
\[
  m = \dfrac{{1098.4}}{8} \\
  m = 137.3 \;
 \]
Therefore, the slope of secant line of $ f\left( x \right) = 6.1{x^2} - 9.1x $ through the points: $ x = 8 $ and $ x = 16 $ is equal to $ m = 137.3 $

Note: 1. The graph the function $ f\left( x \right) = 6.1{x^2} - 9.1x $ given is

2.We should have a better knowledge in the topic of geometry to solve this type of question easily. We should know the slope form of a line when two points on the function are given.
3. Note that we should always give at least two points on the function to find out the slope for the secant line.