At which root does the graph of $f(x)=(x+4)^6(x+7)^5$ cross the $x$-axis?
A. -7
B. -4
C. 4
D. 7
Answer (1)
The roots of the function f ( x ) = ( x + 4 ) 6 ( x + 7 ) 5 are x = − 4 and x = − 7 .
The graph crosses the x-axis at a root if its multiplicity is odd.
The root x = − 4 has multiplicity 6 (even), so the graph does not cross the x-axis there.
The root x = − 7 has multiplicity 5 (odd), so the graph crosses the x-axis there. The final answer is − 7 .
Explanation
Understanding the Problem We are given the function f ( x ) = ( x + 4 ) 6 ( x + 7 ) 5 and asked to find at which root the graph crosses the x-axis.
Finding the Roots The roots of the function are the values of x for which f ( x ) = 0 . Thus, the roots are x = − 4 and x = − 7 .
Understanding Crossing the x-axis The graph of a function crosses the x-axis at a root if the multiplicity of the root is odd. The multiplicity of a root is the exponent of the corresponding factor in the function's equation.
Analyzing the Root x = -4 For the root x = − 4 , the factor is ( x + 4 ) 6 . The exponent is 6, which is even. Therefore, the graph touches the x-axis at x = − 4 but does not cross it.
Analyzing the Root x = -7 For the root x = − 7 , the factor is ( x + 7 ) 5 . The exponent is 5, which is odd. Therefore, the graph crosses the x-axis at x = − 7 .
Conclusion Therefore, the graph of f ( x ) crosses the x-axis at x = − 7 .
Examples
Understanding where a graph crosses the x-axis is useful in many real-world applications. For example, in physics, the x-axis might represent time, and the y-axis might represent the height of a projectile. The points where the graph crosses the x-axis would then represent the times when the projectile hits the ground. In business, the x-axis might represent the number of items sold, and the y-axis might represent profit. The points where the graph crosses the x-axis would then represent the break-even points, where the business is neither making a profit nor losing money. Analyzing the behavior of functions around their roots helps in understanding the underlying phenomena in various fields.
