Which statement describes the graph of [tex]f(x)=-4 x^3-28 x^2-32 x+64[/tex]?
The graph crosses the [tex]x[/tex]-axis at [tex]x=4[/tex] and touches the [tex]x[/tex]-axis at [tex]x=-1[/tex].
The graph touches the [tex]x[/tex]-axis at [tex]x=4[/tex] and crosses the [tex]x[/tex]-axis at [tex]x=-1[/tex].
The graph crosses the [tex]x[/tex]-axis at [tex]x=-4[/tex] and touches the [tex]x[/tex]-axis at [tex]x=1[/tex].
The graph touches the [tex]x[/tex]-axis at [tex]x=-4[/tex] and crosses the [tex]x[/tex]-axis at [tex]x=1[/tex].

The graph of the function touches the x-axis at x = − 4 and crosses the x-axis at x = 1 . This behavior is determined by the multiplicity of the roots of the function. Therefore, the correct option is that the graph touches the x-axis at x = − 4 and crosses at x = 1 .
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Answered by Anonymous | 2025-07-04

Find the roots of the function f ( x ) = − 4 x 3 − 28 x 2 − 32 x + 64 , which are x = − 4 and x = 1 .
Determine the multiplicity of each root: x = − 4 has multiplicity 2, and x = 1 has multiplicity 1.
Since x = − 4 has even multiplicity, the graph touches the x-axis at x = − 4 .
Since x = 1 has odd multiplicity, the graph crosses the x-axis at x = 1 . Therefore, the graph touches the x -axis at x = − 4 and crosses the x -axis at x = 1 . \boxed{The graph touches the x-axis at x=-4 an d crosses t h e x − a x i s a t x=1$.}

Explanation

Problem Analysis We are given the function f ( x ) = − 4 x 3 − 28 x 2 − 32 x + 64 and asked to determine the behavior of its graph at its roots (x-intercepts). Specifically, we need to identify whether the graph crosses or touches the x-axis at each root.

Finding the Roots To find the roots of the function, we set f ( x ) = 0 :
− 4 x 3 − 28 x 2 − 32 x + 64 = 0 We can factor out a -4: − 4 ( x 3 + 7 x 2 + 8 x − 16 ) = 0 This simplifies to: x 3 + 7 x 2 + 8 x − 16 = 0

Identifying the Roots We can use the tool to find the roots of the cubic equation x 3 + 7 x 2 + 8 x − 16 = 0 . The roots are x = − 4 and x = 1 .

Determining Multiplicity To determine the multiplicity of each root, we can use the tool. The multiplicity of the root x = − 4 is 2, and the multiplicity of the root x = 1 is 1.

Analyzing Graph Behavior Now, we analyze the behavior of the graph at each root:



At x = − 4 , the root has a multiplicity of 2, which is even. This means the graph touches the x-axis at x = − 4 .
At x = 1 , the root has a multiplicity of 1, which is odd. This means the graph crosses the x-axis at x = 1 .


Conclusion Therefore, the graph touches the x-axis at x = − 4 and crosses the x-axis at x = 1 .

Examples
Understanding the behavior of polynomial functions at their roots is crucial in various fields. For instance, in engineering, when designing a bridge, engineers need to analyze the stability of the structure. The roots of a polynomial equation can represent critical points where the structure might be vulnerable. If a root has an even multiplicity, it indicates a point of tangency, suggesting a potential weak spot that needs reinforcement. Conversely, a root with odd multiplicity indicates a point where the structure crosses a stable equilibrium, which might require different design considerations. By analyzing the multiplicity of roots, engineers can make informed decisions to ensure the safety and durability of the bridge.

Answered by GinnyAnswer | 2025-07-04