When a factor $x-k$ is raised to an odd power, the graph crosses the $x$-axis at $x=k$.
When a factor $x-k$ is raised to an even power, the graph only touches the $x$-axis at $x=k$.
Describe the graph of the function $g(x)=(x-1)(x+4)^3(x+5)^2$.
The graph:
A. crosses the axis at (1,0).
B. crosses the axis at (-4,0).
C. touches the axis at (-5,0).
Answer (2)
The graph crosses the axis at ( 1 , 0 ) because the factor ( x − 1 ) has an odd power.
The graph crosses the axis at ( − 4 , 0 ) because the factor ( x + 4 ) 3 has an odd power.
The graph touches the axis at ( − 5 , 0 ) because the factor ( x + 5 ) 2 has an even power.
Therefore, the graph crosses the axis at ( 1 , 0 ) and ( − 4 , 0 ) , and touches the axis at ( − 5 , 0 ) . crosses at ( 1 , 0 ) and ( − 4 , 0 ) , touches at ( − 5 , 0 )
Explanation
Understanding the Problem We are given the function g ( x ) = ( x − 1 ) ( x + 4 ) 3 ( x + 5 ) 2 and we need to describe its behavior at the x-intercepts. The key concept here is understanding how the multiplicity of a root affects the graph's behavior at that point.
Analyzing the factor (x-1) The factor ( x − 1 ) has an exponent of 1, which is odd. This means the graph crosses the x-axis at x = 1 . So, the graph crosses the axis at ( 1 , 0 ) .
Analyzing the factor (x+4)^3 The factor ( x + 4 ) 3 has an exponent of 3, which is odd. This means the graph crosses the x-axis at x = − 4 . So, the graph crosses the axis at ( − 4 , 0 ) .
Analyzing the factor (x+5)^2 The factor ( x + 5 ) 2 has an exponent of 2, which is even. This means the graph touches the x-axis at x = − 5 . So, the graph touches the axis at ( − 5 , 0 ) .
Final Answer In summary, the graph crosses the x-axis at ( 1 , 0 ) and ( − 4 , 0 ) , and it touches the x-axis at ( − 5 , 0 ) .
Examples
Understanding the behavior of polynomial functions at their roots is crucial in various fields. For instance, in engineering, when designing a bridge, engineers analyze the function representing the bridge's structure. The roots of the function indicate points of support or stress. Knowing whether the graph crosses or touches the x-axis at these points helps engineers understand the stability and load-bearing capacity of the bridge. Similarly, in economics, understanding the behavior of supply and demand curves at equilibrium points can help predict market trends and make informed decisions.
The graph crosses the x-axis at ( 1 , 0 ) and ( − 4 , 0 ) , and it touches the axis at ( − 5 , 0 ) . Hence, the correct multiple-choice options are A and B for crossing and C for touching the axis. The function's behavior at these points is determined by the odd or even nature of the factors' exponents.
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