To determine which graph could represent the function [tex]\( f(x) = x^3 + 8 \)[/tex], let's analyze its key characteristics and plot some points on the graph:
1. Understand the Function: The function [tex]\( f(x) = x^3 + 8 \)[/tex] is a cubic function. Cubic functions typically have an 'S' shaped curve with one inflection point.
2. Key Characteristics:
- Inflection Point: This is where the curve changes concavity. For cubic functions [tex]\( ax^3 + bx^2 + cx + d \)[/tex], the inflection point typically occurs at the point where the second derivative changes sign.
- Y-intercept: To find where the curve crosses the y-axis, set [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^3 + 8 = 8 \][/tex]
So, the graph will cross the y-axis at [tex]\( (0, 8) \)[/tex].
- Behavior at Large Values: For large positive x, the function behaves like [tex]\( x^3 \)[/tex], so it goes to positive infinity. For large negative x, it also behaves like [tex]\( x^3 \)[/tex], so it goes to negative infinity.
3. Evaluate the Function at Several Points: Let's calculate the function values at a few points to get an idea of the graph's shape.
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = (-2)^3 + 8 = -8 + 8 = 0 \][/tex]
Point: [tex]\((-2, 0)\)[/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = (-1)^3 + 8 = -1 + 8 = 7 \][/tex]
Point: [tex]\((-1, 7)\)[/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^3 + 8 = 8 \][/tex]
Point: [tex]\((0, 8)\)[/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1^3 + 8 = 1 + 8 = 9 \][/tex]
Point: [tex]\((1, 9)\)[/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 2^3 + 8 = 8 + 8 = 16 \][/tex]
Point: [tex]\((2, 16)\)[/tex]
4. Plot the Points:
- [tex]\((-2, 0)\)[/tex]
- [tex]\((-1, 7)\)[/tex]
- [tex]\((0, 8)\)[/tex]
- [tex]\((1, 9)\)[/tex]
- [tex]\((2, 16)\)[/tex]
5. Shape and Traits of the Graph:
- The y-intercept is at [tex]\((0, 8)\)[/tex].
- The graph passes through the points [tex]\((-2, 0)\)[/tex], [tex]\((-1, 7)\)[/tex], [tex]\((0, 8)\)[/tex], [tex]\((1, 9)\)[/tex], and [tex]\((2, 16)\)[/tex].
- The general shape of the cubic function [tex]\( x^3 \)[/tex] dictates an 'S' curve behavior, passing from the third quadrant to the first quadrant through the inflection point.
Based on these computations and characteristics, the graph of [tex]\( f(x) = x^3 + 8 \)[/tex] would show an upward curve passing through the points mentioned, with an 'S' shape that includes these points. The graph should have one inflection point, crossing the y-axis at [tex]\((0, 8)\)[/tex], and showing the behavior as described above. Select the graph that matches this description and plots the points accurately.
