Answer :
- Expand the squared term: $(x+5)^2 = x^2 + 10x + 25$.
- Multiply the expanded term by -9: $-9(x^2 + 10x + 25) = -9x^2 - 90x - 225$.
- Add 4 to the result: $-9x^2 - 90x - 225 + 4 = -9x^2 - 90x - 221$.
- The standard form of the function is $\boxed{f(x)=-9 x^2-90 x-221}$.
### Explanation
1. Understanding the Problem
We are given the function $f(x)=-9(x+5)^2+4$ and we want to find its standard form, which is $f(x) = ax^2 + bx + c$. This means we need to expand and simplify the given expression.
2. Expanding the Square
First, let's expand the squared term $(x+5)^2$. Using the formula $(a+b)^2 = a^2 + 2ab + b^2$, we have $(x+5)^2 = x^2 + 2(5)x + 5^2 = x^2 + 10x + 25$.
3. Multiplying by -9
Next, we multiply the expanded square by -9: $-9(x^2 + 10x + 25) = -9x^2 - 90x - 225$.
4. Adding 4
Finally, we add 4 to the result: $-9x^2 - 90x - 225 + 4 = -9x^2 - 90x - 221$.
5. Final Answer
So, the standard form of the function is $f(x) = -9x^2 - 90x - 221$. Comparing this with the given options, we see that the correct answer is $f(x)=-9 x^2-90 x-221$.
### Examples
Understanding quadratic functions and their standard form is crucial in various fields, such as physics (projectile motion), engineering (designing parabolic reflectors), and economics (modeling cost and revenue). For instance, if you're launching a rocket, the height of the rocket over time can be modeled by a quadratic function. Converting the function to standard form helps in easily identifying key parameters like the maximum height and the time at which it is reached, enabling engineers to optimize the rocket's trajectory.
- Multiply the expanded term by -9: $-9(x^2 + 10x + 25) = -9x^2 - 90x - 225$.
- Add 4 to the result: $-9x^2 - 90x - 225 + 4 = -9x^2 - 90x - 221$.
- The standard form of the function is $\boxed{f(x)=-9 x^2-90 x-221}$.
### Explanation
1. Understanding the Problem
We are given the function $f(x)=-9(x+5)^2+4$ and we want to find its standard form, which is $f(x) = ax^2 + bx + c$. This means we need to expand and simplify the given expression.
2. Expanding the Square
First, let's expand the squared term $(x+5)^2$. Using the formula $(a+b)^2 = a^2 + 2ab + b^2$, we have $(x+5)^2 = x^2 + 2(5)x + 5^2 = x^2 + 10x + 25$.
3. Multiplying by -9
Next, we multiply the expanded square by -9: $-9(x^2 + 10x + 25) = -9x^2 - 90x - 225$.
4. Adding 4
Finally, we add 4 to the result: $-9x^2 - 90x - 225 + 4 = -9x^2 - 90x - 221$.
5. Final Answer
So, the standard form of the function is $f(x) = -9x^2 - 90x - 221$. Comparing this with the given options, we see that the correct answer is $f(x)=-9 x^2-90 x-221$.
### Examples
Understanding quadratic functions and their standard form is crucial in various fields, such as physics (projectile motion), engineering (designing parabolic reflectors), and economics (modeling cost and revenue). For instance, if you're launching a rocket, the height of the rocket over time can be modeled by a quadratic function. Converting the function to standard form helps in easily identifying key parameters like the maximum height and the time at which it is reached, enabling engineers to optimize the rocket's trajectory.
