Evaluate the function 

f left parenthesis x right parenthesis equals negative 3 left parenthesis x minus 4 right parenthesis squared left parenthesis x plus 2 right parenthesisf(x)=−3(x−4)2(x+2) 

at the indicated values.

a. 

​f(00​)

b. 

​f(33​)

Question content area bottom

Part 1

a. 

​f(00​)equals=enter your response here  

​(Type an integer or a simplified​ fraction.)

Part 2

b. 

​f(33​)equals=enter your response here  

​(Type an integer or a simplified​ fraction.)

To evaluate the function \( f(x) = -3(x - 4)^2(x + 2) \) at the indicated values, we’ll substitute \( x = 0 \) and \( x = 3 \) into the function and simplify.

### Part 1: \( f(0) \)
Substitute \( x = 0 \) into the function:
\[
f(0) = -3(0 - 4)^2(0 + 2)
\]
Simplify each part inside the parentheses:
\[
= -3(-4)^2 \cdot 2
\]
\[
= -3 \cdot 16 \cdot 2
\]
\[
= -3 \cdot 32 = -96
\]
So, \( f(0) = -96 \).

### Part 2: \( f(3) \)
Substitute \( x = 3 \) into the function:
\[
f(3) = -3(3 - 4)^2(3 + 2)
\]
Simplify each part inside the parentheses:
\[
= -3(-1)^2 \cdot 5
\]
\[
= -3 \cdot 1 \cdot 5
\]
\[
= -3 \cdot 5 = -15
\]
So, \( f(3) = -15 \).

### Final Answers
- \( f(0) = -96 \)
- \( f(3) = -15 \)

Would you like further details on any part of this solution?

Here are some related questions to deepen understanding:
1. What happens to \( f(x) \) when \( x = 4 \) or \( x = -2 \)?
2. How would the function change if the coefficient \(-3\) were positive?
3. How does the function behave as \( x \) approaches very large positive or negative values?
4. What are the roots of \( f(x) \) when set to zero?
5. How does squaring \( (x - 4) \) affect the function's graph shape?

**Tip:** When evaluating functions, simplify each term step-by-step for clarity and accuracy.

Evaluate the function f(x) = -3(x - 4)^2(x + 2) at the indicated values. a. f(0), b. f(3).

Learn how to evaluate the polynomial function f(x) = -3(x - 4)^2(x + 2) by substituting specific values for x. This step-by-step solution simplifies expressions and covers polynomial function properties, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation