https://file.aiquickdraw.com/mathbot/file/17266923720856fytuiwd.jpg

**Insert** an algebraic expression for the function with independent variable value of opposite sign **here. f_1=? and Insert** an algebraic expression for the function multiplied by factor of -1 **here. f_2=?**

The function provided in the graph is:

\[
f(x) = 5x^3 + 3
\]

Now, let's address your two questions:

1. **Find an algebraic expression for the function with the independent variable's value having the opposite sign (i.e., for \( f(-x) \)):**

   Substituting \( -x \) into the function:

   \[
   f_1(x) = f(-x) = 5(-x)^3 + 3 = 5(-x^3) + 3 = -5x^3 + 3
   \]

   So, the expression for \( f_1(x) \) is:

   \[
   f_1(x) = -5x^3 + 3
   \]

2. **Find an algebraic expression for the function multiplied by a factor of -1 (i.e., for \( -f(x) \)):**

   Multiplying the entire function \( f(x) \) by -1:

   \[
   f_2(x) = -f(x) = -(5x^3 + 3) = -5x^3 - 3
   \]

   So, the expression for \( f_2(x) \) is:

   \[
   f_2(x) = -5x^3 - 3
   \]

Would you like more details or further explanations?

### Related Questions:
1. How does the graph of \( f(x) \) change when the input variable is negated?
2. What does multiplying a function by -1 do to its graph?
3. What is the effect on symmetry when negating the input variable in cubic functions?
4. How do odd and even powers in polynomials behave when you negate the input variable?
5. What other transformations can you apply to \( f(x) = 5x^3 + 3 \)?

**Tip:** Negating the variable reflects the graph across the y-axis, while multiplying the function by -1 reflects it across the x-axis.

Insert an algebraic expression for the function with independent variable value of opposite sign here. f_1=? and Insert an algebraic expression for the function multiplied by factor of -1 here. f_2=?

Understand how to find algebraic expressions for f_1(x) and f_2(x) for the cubic function f(x) = 5x^3 + 3. This solution includes negating the variable and multiplying the function by -1, which are key concepts in algebraic graph transformations. Suitable for Grades 10-12.

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