We can see two functions and we have to find the average rate of change of them from x = 1 to x = 2.
The average rate of change is given by the formula:
[tex]\text{ The average rate of change}=\frac{\text{ change in y}}{\text{ change in x}}=\frac{y_2-y_1}{x_2-x_1}=\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]Therefore, to find the average rate of change of a function, we have to find the values of the function for the given values, x1 = 1, x2 = 2, and then find the corresponding ratio as follows:
To find it, we can proceed as follows:
1. Find the value of the function for x1 = 1 and x2 = 2:
[tex]\begin{gathered} f(x_2)=f(2)=2(2)-5=4-5=-1 \\ \\ f(x_1)=f(1)=2(1)-5=2-5=-3 \end{gathered}[/tex]2. Then the average rate of change is:
[tex]A_{rateofchange(1\text{ to 2\rparen}}=\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{-1-(-3)}{2-1}=\frac{-1+3}{1}=2[/tex]Therefore, the average rate of change for f(x) = 2x - 5 from x = 1 to x = 2 is 2.
1. We can proceed in a similar way as before to find the average rate of change in this case:
[tex]\begin{gathered} f(x)=-3x^2+5x-1 \\ \\ f(x_2)=f(2)=-3(2)^2+5(2)-1=-3(4)+10-1=-12+10-1=-2-1 \\ \\ f(x_2)=-3 \end{gathered}[/tex]2. And
[tex]\begin{gathered} f(x_1)=f(1)=-3(1)^2+5(1)-1=-3(1)+5-1=-3+5-1=2-1=1 \\ \\ f(x_1)=1 \end{gathered}[/tex]3. Then the average rate of change of this function from x = 1 to x = 2 is:
[tex]\begin{gathered} A_{\text{ rate of change\lparen x =1, x=2\rparen}}=\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{-3-1}{2-1}=\frac{-4}{1}=-4 \\ \\ A_{\text{ rate of change\lparen x =1, x=2\rparen}}=-4 \end{gathered}[/tex]Therefore, in summary, we can conclude that the average rate of change from x = 1 to x = 2 is:
1. For the function f(x) = 2x - 5 is 2
2.For the function f(x) = -3x²+5x-1 is -4