5. A civil anglnoer is designing the cross-section of a new support beam for a bridge. The initial design is modolad by the function $$ f(x)=x^{2} $$, where $$ x $$ reprosonts the horizontal distance from the center of the boam and $$ f(x) $$ reprosents the height of the boam at that point. However, after struclural analysig, it's determined that tha beam needs to be even wider to bettor distribute the load. Explain how tho enginear can change the function to achieves this wider beam deaign. Answer: the can make the value of b lower so that it compresses,

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1. The initial design is modeled by 
   $$
   f(x)=x^2.
   $$
   This parabola rises too steeply if the beam is to be wide.

2. A wider beam design requires a parabola that “opens up” more, meaning it is less steep. One way to achieve this is by horizontally stretching the graph.

3. A horizontal stretch by a factor of $$ k > 1 $$ replaces $$ x $$ with $$\frac{x}{k}$$. That is, define the new function by
   $$
   g(x)=\left(\frac{x}{k}\right)^2.
   $$
   Here, as $$ k $$ increases, the parabola becomes wider.

4. Equivalently, the function can be written as
   $$
   g(x)=\frac{1}{k^2}x^2.
   $$
   Choosing a suitable value of $$ k $$ (with $$ k>1 $$) will give the engineer the required beam width.

5. In summary, to achieve the wider beam design the engineer can use the transformation
   $$
   x \to \frac{x}{k} \quad \text{(with } k>1\text{)},
   $$
   which gives the new design function 
   $$
   g(x)=\left(\frac{x}{k}\right)^2.
   $$
To make the beam wider, the engineer can use a horizontal stretch by increasing the value of $$ k $$ in the function $$ g(x) = \left(\frac{x}{k}\right)^2 $$, where $$ k > 1 $$. This will make the parabola flatter and the beam wider.

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