7. evaluate the difference quotient, $\frac{f(a + h)-f(a)}{h}$, for the function $f(x)=x^{2}-4x + 3$

Explanation:

Step1: Find $f(a + h)$

Substitute $x=a + h$ into $f(x)=x^{2}-4x + 3$. $f(a + h)=(a + h)^{2}-4(a + h)+3=a^{2}+2ah+h^{2}-4a-4h + 3$

Step2: Find $f(a)$

Substitute $x = a$ into $f(x)=x^{2}-4x + 3$. $f(a)=a^{2}-4a + 3$

Step3: Calculate $f(a + h)-f(a)$

$f(a + h)-f(a)=(a^{2}+2ah+h^{2}-4a-4h + 3)-(a^{2}-4a + 3)$ $=a^{2}+2ah+h^{2}-4a-4h + 3 - a^{2}+4a - 3=2ah+h^{2}-4h$

Step4: Calculate the difference quotient

$\frac{f(a + h)-f(a)}{h}=\frac{2ah+h^{2}-4h}{h}$ Factor out $h$ from the numerator: $\frac{h(2a + h-4)}{h}=2a+h - 4$

Answer:

$2a+h - 4$