To determine which graph could be the graph of the function [tex]\( f(x) = x^3 + 8 \)[/tex], we need to understand the shape and behavior of this function. Let's analyze it step-by-step and look at some specific values to get a clearer picture:
1. General Shape of the Function:
- The function [tex]\( f(x) = x^3 + 8 \)[/tex] is a cubic function.
- Cubic functions typically have an S-shaped curve.
- This particular function is shifted 8 units upward due to the [tex]\( +8 \)[/tex].
2. Key Characteristics:
- Cubic functions are defined for all real numbers [tex]\(-\infty < x < \infty\)[/tex].
- [tex]\( f(x) \)[/tex] is increasing for all [tex]\( x \)[/tex] because the derivative [tex]\( 3x^2 \)[/tex] is always non-negative.
- There are no local maxima or minima in this function since it is always increasing.
3. Specific Points:
Let's calculate some specific points to help us match the graph.
- For [tex]\( x = -10 \)[/tex], [tex]\( f(-10) = (-10)^3 + 8 = -1000 + 8 = -992 \)[/tex]
- For [tex]\( x = -3 \)[/tex], [tex]\( f(-3) = (-3)^3 + 8 = -27 + 8 = -19 \)[/tex]
- For [tex]\( x = -2 \)[/tex], [tex]\( f(-2) = (-2)^3 + 8 = -8 + 8 = 0 \)[/tex]
- For [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0^3 + 8 = 8 \)[/tex]
- For [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 1^3 + 8 = 1 + 8 = 9 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 2^3 + 8 = 8 + 8 = 16 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( f(3) = 3^3 + 8 = 27 + 8 = 35 \)[/tex]
- For [tex]\( x = 10 \)[/tex], [tex]\( f(10) = 10^3 + 8 = 1000 + 8 = 1008 \)[/tex]
4. Behavior at Infinity:
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) = x^3 + 8 \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) = x^3 + 8 \to -\infty \)[/tex].
Now, with these points and behaviors in mind, we can match these features against the given graphs:
- Look for an S-shaped curve.
- It should pass through the specific points we calculated.
Given:
- The y-values are shifting from negative very large to positive very large symmetrically but not linearly.
The correct graph will:
- Cross the y-axis at [tex]\( (0, 8) \)[/tex]
- Keep increasing before and after crossing the y-axis
- Show the S-shaped nature prominently.
