To determine the correct statement about the factor \( k \), we first need to analyze the two U-shaped graphs (which are parabolas) based on the given points.
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Solid Line Parabola: It passes through the points (0, -3), (-1, 0), and (1, 0). The vertex is at (0, -3) and it opens upwards. The x-intercepts are at (-1, 0) and (1, 0).
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Dotted Line Parabola: It passes through the points (0, -3), (-2, 0), and (2, 0). The vertex is also at (0, -3) and it opens upwards. The x-intercepts are at (-2, 0) and (2, 0).
Let's now analyze the relationships between these two parabolas:
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The horizontal distance between the x-intercepts of the solid line is 1 unit each from the origin, whereas for the dotted line, the x-intercepts are 2 units from the origin.
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In effect, the dotted line is a horizontal stretch of the solid line. Specifically, when a parabola is stretched horizontally by a factor \( k \), the distance between the x-intercepts will become \( k \) times the original distance.
Given the x-intercepts for the solid line are at \( x = -1 \) and \( x = 1 \) (1 unit apart), and for the dotted line at \( x = -2 \) and \( x = 2 \) (2 units apart), this indicates:
\[ k \cdot 1 = 2 \Rightarrow k = 2 \]
Thus, it confirms that the dotted line is a horizontal stretch of the solid line with a factor of \( k = 2\).
In conclusion, the correct statement is:
Horizontal stretch with a factor \( k = 2\).
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