Find the difference quotient
mula1" title="\frac{f(x + h) - f(x)}{h}" alt="\frac{f(x + h) - f(x)}{h}" align="absmiddle" class="latex-formula"> for the following function.
![f(x) = -x^2 -3x + 1](https://tex.z-dn.net/?f=f%28x%29%20%3D%20-x%5E2%20-3x%20%2B%201)
2 answers:
Answer:
![-2x - h - 3](https://tex.z-dn.net/?f=-2x%20-%20h%20-%203)
Step-by-step explanation:
Step 1: Define
Difference Quotient: ![\frac{f(x+h)-f(x)}{h}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D)
f(x) = -x² - 3x + 1
f(x + h) means that x = (x + h)
f(x) is just the normal function
Step 2: Find difference quotient
- <u>Substitute:</u>
![\frac{[-(x+h)^2-3(x+h)+1]-(-x^2-3x+1)}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B%5B-%28x%2Bh%29%5E2-3%28x%2Bh%29%2B1%5D-%28-x%5E2-3x%2B1%29%7D%7Bh%7D)
- <u>Expand and Distribute:</u>
![\frac{[-(x^2+2hx+h^2)-3x-3h+1]+x^2+3x-1}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B%5B-%28x%5E2%2B2hx%2Bh%5E2%29-3x-3h%2B1%5D%2Bx%5E2%2B3x-1%7D%7Bh%7D)
- <u>Distribute:</u>
![\frac{-x^2-2hx-h^2-3x-3h+1+x^2+3x-1}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B-x%5E2-2hx-h%5E2-3x-3h%2B1%2Bx%5E2%2B3x-1%7D%7Bh%7D)
- <u>Combine like terms:</u>
![\frac{-2hx-h^2-3h}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B-2hx-h%5E2-3h%7D%7Bh%7D)
- <u>Factor out </u><em><u>h</u></em><u>:</u>
![\frac{h(-2x-h-3)}{h}](https://tex.z-dn.net/?f=%5Cfrac%7Bh%28-2x-h-3%29%7D%7Bh%7D)
- <u>Simplify:</u>
![-2x - h - 3](https://tex.z-dn.net/?f=-2x%20-%20h%20-%203)
Hey there!
We know that formula for the difference quotient is [
]
We need to find f(x + h), so use (x + h) instead of x.
f(x + h) → -(x + h)² because we are given [ -x² ]
-3(h + x) + 1 because we are given [ -3x + 1 ]
Put what we found into the formula and simplify.
![\frac{(-(h+x)^2-3(h+x)+1-(x^2-3x+1)}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B%28-%28h%2Bx%29%5E2-3%28h%2Bx%29%2B1-%28x%5E2-3x%2B1%29%7D%7Bh%7D)
![-h-2x-3](https://tex.z-dn.net/?f=-h-2x-3)
Therefore, the difference quotient for the given function is [ -h - 2x - 3 ]
Best of Luck!
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