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Question: How do you find the slope of the secant line of \( f\left( x \right) = 6.1{x^2} - 9.1x \) through th...

How do you find the slope of the secant line of f(x)=6.1x2−9.1xf\left( x \right) = 6.1{x^2} - 9.1x through the points: x=8x = 8 and x=16x = 16

Explanation

Solution

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=ΔyΔx=y2−y1x2−x1slope = m = \dfrac{{\Delta y}}{{\Delta x}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}. x1,x2{x_1},{x_2} will the values of xx given in the question and y1,y2{y_1},{y_2} are the values of function obtained by putting x1,x2{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 xx as f(x)=6.1x2−9.1xf\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=8x = 8 and x=16x = 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(x)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=ΔyΔx=y2−y1x2−x1=f(x2)−f(x1)x2−x1slope = 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 x1,x2{x_1},{x_2} as 8,168,16 respectively.
Now let’s find out the value for f(x1)f\left( {{x_1}} \right) by substituting all the occurrences of variable xx with the value of x1{x_1} , we have
f(x1)=f(8) ⇒f(8)=(6.1)(8)2−(9.1)(8) ⇒f(8)=390.4−72.8 ⇒f(8)=317.6   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(x2)f\left( {{x_2}} \right) , replacing all the xx with value of x2=16{x_2} = 16 in the function
f(x2)=f(16) ⇒f(16)=(6.1)(16)2−(9.1)(16) ⇒f(16)=1561.6−145.6 ⇒f(16)=1416   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(x1)=317.6f\left( {{x_1}} \right) = 317.6 and f(x2)=1416f\left( {{x_2}} \right) = 1416 .
Now putting these values in the slope of the secant line, we get the slope as

slope=m=y2−y1x2−x1=f(x2)−f(x1)x2−x1 m=1416−317.616−8   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=1098.48 m=137.3   m = \dfrac{{1098.4}}{8} \\\ m = 137.3 \;

Therefore, the slope of secant line of f(x)=6.1x2−9.1xf\left( x \right) = 6.1{x^2} - 9.1x through the points: x=8x = 8 and x=16x = 16 is equal to m=137.3m = 137.3

Note : 1. The graph the function f(x)=6.1x2−9.1xf\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.