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I'm working on an audio synthesis project and I need an equation that can create these lines:

That is, I would like an equation that describes a line that always passes through (0,1) and (1,0), and has a single coefficient parameter that can be varied to create functions of intermediate concavity/convexity like the red line shown.

The concavity/convexity should be symmetric as if mirrored along the line of y=x between the boundaries of (0,1) and (1,0). Anything outside that range is irrelevant.

Is there an equation that can do this?

That is, I would like an equation that describes a line that always passes through (0,1) and (1,0), and has a single coefficient parameter that can be varied to create functions of intermediate concavity/convexity like the red line shown.

The concavity/convexity should be symmetric as if mirrored along the line of y=x between the boundaries of (0,1) and (1,0). Anything outside that range is irrelevant.

Is there an equation that can do this?

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