question 7 of 13, step 1 of 1 correct given the following function, determine the difference quotient, (f(x…

Explanation:

Step1: Find (f(x + k))

Substitute (x + k) into (f(x)): [ \begin{align*} f(x + k)&=4(x + k)^2-8(x + k)+6\ &=4(x^{2}+2kx + k^{2})-8x-8k + 6\ &=4x^{2}+8kx+4k^{2}-8x - 8k+6 \end{align*} ]

Step2: Calculate (f(x + k)-f(x))

[ \begin{align*} f(x + k)-f(x)&=(4x^{2}+8kx + 4k^{2}-8x-8k + 6)-(4x^{2}-8x + 6)\ &=4x^{2}+8kx+4k^{2}-8x-8k + 6 - 4x^{2}+8x - 6\ &=8kx+4k^{2}-8k \end{align*} ]

Step3: Calculate the difference - quotient (\frac{f(x + k)-f(x)}{k})

[ \begin{align*} \frac{f(x + k)-f(x)}{k}&=\frac{8kx+4k^{2}-8k}{k}\ &=\frac{k(8x + 4k-8)}{k}\ &=8x+4k - 8 \end{align*} ]

Answer:

(8x + 4k-8)