Question

Find and simplify the difference quotient for the function f(x)=−5x²+2x+9.

Answer 1

The difference quotient of the function f(x) = -5x² + 2x + 9 is found by substituting x + h into the function, expanding, and simplifying. The simplified difference quotient formula is (f(x + h) - f(x)) / h, which results in -10x - 5h + 2.

Step-by-step explanation:

To find and simplify the difference quotient of the function f(x) = −5x² + 2x + 9, we use the formula for the difference quotient, which is (f(x + h) - f(x)) / h. This requires us to first calculate f(x + h). Let's do this step by step:

1. Substitute x + h into the function: f(x + h) = −5(x + h)² + 2(x + h) + 9.
2. Expand the square and distribute: f(x + h) = −5(x² + 2xh + h²) + 2x + 2h + 9.
3. Simplify the function: f(x + h) = −5x² - 10xh - 5h² + 2x + 2h + 9.
4. Find the difference: f(x + h) - f(x) = (−5x² - 10xh - 5h² + 2x + 2h + 9) - (−5x² + 2x + 9).
5. Simplify the difference: f(x + h) - f(x) = -10xh - 5h² + 2h.
6. Divide by h to find the difference quotient: (-10xh - 5h² + 2h) / h = -10x - 5h + 2.

In conclusion, the simplified difference quotient for the function f(x) is -10x - 5h + 2.