How does the graph of the catenary y = a cosh x/a change as a > 0 increases?

Verified step by step guidance

1. Recall the definition of the catenary function: \(y = a \cosh\left(\frac{x}{a}\right)\), where \(a > 0\) is a parameter that affects the shape of the curve.
2. Understand that \(\cosh(z)\) is the hyperbolic cosine function, which is always positive and symmetric about the \(y\)-axis, with a minimum value of 1 at \(z=0\).
3. Analyze how changing \(a\) affects the graph: since \(a\) appears both as a vertical scaling factor and inside the argument of \(\cosh\), increasing \(a\) will stretch the graph vertically and horizontally.
4. More specifically, as \(a\) increases, the curve becomes wider (flatter) because the argument \(\frac{x}{a}\) changes more slowly with \(x\), and the minimum value at \(x=0\) increases proportionally to \(a\).
5. Summarize that increasing \(a\) results in a catenary that is less steep near the vertex and has a higher minimum point, making the curve appear broader and taller.

Key Concepts

Definition and Properties of the Catenary

The catenary is the curve formed by a hanging flexible chain or cable under its own weight, described by the equation y = a cosh(x/a). It is distinct from a parabola and characterized by its hyperbolic cosine shape, symmetric about the y-axis, with the parameter 'a' controlling its steepness and width.

Effect of the Parameter 'a' on the Graph

In the equation y = a cosh(x/a), the parameter 'a' scales both the vertical and horizontal dimensions of the curve. As 'a' increases, the catenary becomes wider and flatter near the vertex, meaning the curve stretches horizontally and the minimum point rises, reflecting a less steep shape.

Hyperbolic Cosine Function Behavior

The hyperbolic cosine function, cosh(t), grows exponentially for large |t| and has a minimum value of 1 at t = 0. Understanding cosh's shape helps explain the catenary's form: it is always positive, symmetric, and smooth, with the parameter 'a' affecting the input scaling, thus altering the curve's width and height.