If f(x) = -x² - 2x - 6, then f(2) - f(t) =
Understand the Problem
The question requires calculating the difference between the function values at two points: f(2) and f(t) for the given function f(x) = -x² - 2x - 6. To solve this, we will first compute f(2) and then express f(t) in terms of t before finding the difference.
Answer
$$ f(2) - f(t) = t^2 + 2t - 8 $$
Answer for screen readers
The final result is: $$ f(2) - f(t) = t^2 + 2t - 8 $$
Steps to Solve
-
Calculate f(2)
To find $f(2)$, substitute $x = 2$ into the function $f(x) = -x^2 - 2x - 6$. $$ f(2) = -2^2 - 2(2) - 6 $$ -
Simplify f(2)
Now, simplify the expression: $$ f(2) = -4 - 4 - 6 = -14 $$ -
Express f(t)
Next, we need to express $f(t)$ by substituting $x = t$ into the function: $$ f(t) = -t^2 - 2t - 6 $$ -
Find the difference f(2) - f(t)
Now we find the difference between $f(2)$ and $f(t)$: $$ f(2) - f(t) = -14 - (-t^2 - 2t - 6) $$ -
Simplify the expression
Distributing the negative sign and simplifying: $$ f(2) - f(t) = -14 + t^2 + 2t + 6 $$
$$ = t^2 + 2t - 8 $$
The final result is: $$ f(2) - f(t) = t^2 + 2t - 8 $$
More Information
This expression represents the difference between the function values at the specified points. The function $f(x) = -x^2 - 2x - 6$ is quadratic, indicating it has a parabolic shape. The result can be interpreted as how the function changes from $x = 2$ to $x = t$.
Tips
- Forgetting to distribute the negative sign when finding $f(2) - f(t)$.
- Miscalculating the value of $f(2)$; double-check each arithmetic step.
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