find the average rate of change of f(x)=4x^2 - 6 on the interval 1,a. your answer will be an expression…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[x_1,x_2]$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. Here, $x_1 = 1$, $x_2=a$, and $f(x)=4x^{2}-6$.

Step2: Find $f(1)$ and $f(a)$

First, find $f(1)$: Substitute $x = 1$ into $f(x)$: $f(1)=4\times1^{2}-6=4 - 6=-2$. Then, find $f(a)$: Substitute $x = a$ into $f(x)$: $f(a)=4a^{2}-6$.

Step3: Calculate the average rate of change

Substitute $f(1)$ and $f(a)$ into the average - rate - of - change formula: $\frac{f(a)-f(1)}{a - 1}=\frac{(4a^{2}-6)-(-2)}{a - 1}=\frac{4a^{2}-6 + 2}{a - 1}=\frac{4a^{2}-4}{a - 1}$. Factor the numerator: $\frac{4(a^{2}-1)}{a - 1}$. Since $a^{2}-1=(a + 1)(a - 1)$ (using the difference - of - squares formula $x^{2}-y^{2}=(x + y)(x - y)$ with $x=a$ and $y = 1$), we have $\frac{4(a + 1)(a - 1)}{a - 1}$. Cancel out the common factor $(a - 1)$ (assuming $a\neq1$): $4(a + 1)$.

Answer:

$4(a + 1)$