According to the theorem of parallel axes `I = I_("cm") + Mx^(2)`, the graph between I and x will be

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Answer

To solve the problem regarding the graph between moment of inertia \( I \) and distance \( x \) according to the theorem of parallel axes, we can follow these steps:

### Step 1: Understand the Theorem of Parallel Axes
The theorem states that the moment of inertia \( I \) about any axis parallel to an axis through the center of mass (CM) is given by:
\[ I = I_{\text{cm}} + Mx^2 \]
where:
- \( I \) is the moment of inertia about the new axis,
- \( I_{\text{cm}} \) is the moment of inertia about the center of mass axis,
- \( M \) is the mass of the body,
- \( x \) is the distance between the two parallel axes.

### Step 2: Analyze the Equation
From the equation, we can see that \( I \) is a function of \( x^2 \). This indicates that the relationship between \( I \) and \( x \) is quadratic.

### Step 3: Determine the Value of \( I \) at \( x = 0 \)
When \( x = 0 \):
\[ I = I_{\text{cm}} + M(0)^2 = I_{\text{cm}} \]
This means that the moment of inertia \( I \) at \( x = 0 \) is equal to \( I_{\text{cm}} \), which is a non-zero constant.

### Step 4: Behavior of \( I \) as \( x \) Increases
As \( x \) increases (either positively or negatively), the term \( Mx^2 \) increases, leading to an increase in \( I \). Thus, \( I \) increases as \( |x| \) increases.

### Step 5: Graphical Representation
Since \( I \) is proportional to \( x^2 \), the graph of \( I \) versus \( x \) will be a parabola opening upwards. The vertex of this parabola is at the point where \( x = 0 \) and \( I = I_{\text{cm}} \).

### Step 6: Conclusion about the Graph
The graph will not intersect the x-axis (since \( I \) is never zero for \( x = 0 \)), and it will be symmetric about the y-axis because \( I \) depends on \( x^2 \). Therefore, the graph will extend upwards on both sides of the origin.

### Final Answer
The graph between \( I \) and \( x \) will be a parabola that opens upwards, with its vertex at \( (0, I_{\text{cm}}) \).

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