\begin{tabular}{|l} 5) \\ A line is dilated by a scale factor of \( \frac{1}{3} \) centered at a \\ point on the line. Which statement is correct about \\ the image of the line? \\ 1) Its slope is changed by a scale factor of \( \frac{1}{3} \), \\ 2) Its \( y \)-intercept is changed by a scale factor of \( \frac{1}{3} \), \\ 3) Its slope and \( y \)-intercept are changed by a scale \\ factor of \( \frac{1}{3} \). \\ 4) Theimage of the line and the pre-image are the \\ same line. \end{tabular}

When a line undergoes dilation centered at a point on that line, the entire line transforms while keeping its orientation intact. Since the dilation maintains the direction and simply reduces or enlarges the dimensions, the slope of the line remains unchanged. However, the y-intercept isn't transformed proportionally by the scale factor; it can actually be altered and won't maintain the same relationship as if it were a geometrical figure. So, the correct statement is that the image of the line and the pre-image are the same line. In real-life applications, dilations like these can help in fields such as graphic design and architecture, where maintaining proportions and orientations is crucial while resizing images or blueprints. Understanding the properties of dilated shapes allows designers to create visually appealing and coherent designs, ensuring that the intended message or form remains intact even when scaled up or down.