Question

solve the function at that value to determine if the graph is above or below x-axis.

X^3 + 9x^2 + 23x + 15

Answer 1

To determine whether the graph of the cubic function \( f(x) = x^3 + 9x^2 + 23x + 15 \) is above or below the x-axis, you can evaluate the function at specific points to see where it is positive (above the x-axis) or negative (below the x-axis).

We can start by checking some values for \( x \).

1. **Calculate the value of \( f(x) \) at \( x = -5 \):**
\[
f(-5) = (-5)^3 + 9(-5)^2 + 23(-5) + 15
\]
\[
= -125 + 225 - 115 + 15
\]
\[
= -125 + 225 - 115 + 15 = -125 + 225 - 100 = 0
\]

Thus, \( f(-5) = 0 \). This means the graph intersects the x-axis at \( x = -5 \).

2. **Now, check \( f(x) \) for values before and after \( -5 \) to find if it is above or below the x-axis.**

- For \( x = -6 \):
\[
f(-6) = (-6)^3 + 9(-6)^2 + 23(-6) + 15
\]
\[
= -216 + 324 - 138 + 15
\]
\[
= -216 + 324 - 138 + 15 = -216 + 324 - 123 = -15
\]

Thus, \( f(-6) < 0 \) (below the x-axis).

- For \( x = -4 \):
\[
f(-4) = (-4)^3 + 9(-4)^2 + 23(-4) + 15
\]
\[
= -64 + 144 - 92 + 15
\]
\[
= -64 + 144 - 92 + 15 = -64 + 144 - 77 = 3
\]

Thus, \( f(-4) > 0 \) (above the x-axis).

From these calculations, we can conclude the following:

- For \( x < -5 \) (specifically at \( x = -6 \)), the function is below the x-axis.
- At \( x = -5 \), the function intersects the x-axis.
- For \( x > -5 \) (specifically at \( x = -4 \)), the function is above the x-axis.

The behavior of the function suggests that it crosses the x-axis at \( x = -5 \) and is above the x-axis for \( x > -5 \).

In conclusion, the graph of the function \( f(x) = x^3 + 9x^2 + 23x + 15 \) is below the x-axis for values less than \( -5 \) and above the x-axis for values greater than \( -5 \).