Which of the following graphs could be the graph of the function [tex]$f(x)=x^3+8$[/tex]?
Answer (2)
The function has a y-intercept at ( 0 , 8 ) .
The function has a general 'S' shape, typical of cubic functions.
As x approaches positive infinity, f ( x ) approaches positive infinity; as x approaches negative infinity, f ( x ) approaches negative infinity.
The function has a root at x = − 2 .
Therefore, the graph that satisfies these conditions is f ( x ) = x 3 + 8 .
Explanation
Analyzing the Function We are given the function f ( x ) = x 3 + 8 and we need to identify its graph. Let's analyze the properties of this function to determine which graph it corresponds to.
Finding the y-intercept First, let's find the y-intercept of the function. To do this, we set x = 0 and evaluate f ( 0 ) : f ( 0 ) = ( 0 ) 3 + 8 = 0 + 8 = 8 So, the y-intercept is at the point ( 0 , 8 ) . This means the graph of the function must pass through the point ( 0 , 8 ) .
Understanding the Shape Next, let's consider the general shape of the cubic function x 3 . It's an increasing function that passes through the origin ( 0 , 0 ) . The function f ( x ) = x 3 + 8 is a vertical translation of the basic cubic function x 3 by 8 units upwards. Therefore, it will have the same general 'S' shape, but shifted upwards.
Analyzing End Behavior Now, let's analyze the end behavior of the function. As x approaches positive infinity, x 3 approaches positive infinity, and thus f ( x ) = x 3 + 8 also approaches positive infinity. Similarly, as x approaches negative infinity, x 3 approaches negative infinity, and thus f ( x ) = x 3 + 8 also approaches negative infinity.
Finding the Root The function has a root when f ( x ) = 0 . So we need to solve the equation x 3 + 8 = 0 . This gives us x 3 = − 8 . Taking the cube root of both sides, we get x = − 2 . So the function has a real root at x = − 2 , which means the graph passes through the point ( − 2 , 0 ) .
Summary of Characteristics Based on our analysis, the graph of f ( x ) = x 3 + 8 should have the following characteristics:
y-intercept at ( 0 , 8 )
General 'S' shape of a cubic function
End behavior: As x → ∞ , f ( x ) → ∞ , and as x → − ∞ , f ( x ) → − ∞
Root at x = − 2
Therefore, we need to look for a graph that satisfies all these conditions.
Examples
Cubic functions like f ( x ) = x 3 + 8 are used in various real-world applications. For example, they can model population growth, the volume of a cube as a function of its side length, or the trajectory of a projectile. Understanding the shape and properties of cubic functions helps in predicting and analyzing these phenomena.
The graph of the function f ( x ) = x 3 + 8 has a y-intercept at ( 0 , 8 ) , has an 'S' shape, and crosses the x-axis at ( − 2 , 0 ) . It extends downward to the left and upward to the right. Look for a graph that displays these characteristics.
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