NSW Y11 Maths - Extension 1 Functions Graphing-Adding Ordinates

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Given the graphs of \(y = f(x)\)  and \(y = g(x)\) then the graph of \(y = f(x) + g(x)\) can be obtained by adding the ordinates of each \(x\) value.

An interesting example is the cosh function.

If \(f(x) = \dfrac{1}{2}{e^x}\) and \(g(x) = \dfrac{1}{2}{e^{ - x}}\) then \(y = f(x) + g(x)\,\, \to y = \dfrac{1}{2}\left( {{e^x} + {e^{ - x}}} \right)\) which is the cosh or ‘hanging cable’ function.

By subtracting ordinates: \(y = f(x) - g(x)\,\, \to y = \dfrac{1}{2}\left( {{e^x} - {e^{ - x}}} \right)\) this the \(sinh\) function.

Both are known as hyperbolic functions, and yes there is also a \(tanh\) function \(y = \dfrac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}}\).

NSW Syllabus Reference: ME-F1.1: Graphical relationships. This will require student to 

• examine the relationship between the graph of \(y=f(x)\) and the graph of \(y=\dfrac{1}{f(x)}\) and hence sketch the graphs (ACMSM099)
• examine the relationship between the graph of \(y=f(x)\) and the graphs of \(y^2=f(x)\) and \(y=\sqrt{f(x)}\) and hence sketch the graphs
• examine the relationship between the graph of \(y=f(x)\) and the graphs of \(y=|f(x)|\) and \(y=f(x)+g(x)\) and hence sketch the graphs (ACMSM099)
• examine the relationship between the graphs of \(y=f(x)\) and \(y=g(x)\) and the graphs of \(y=f(x)+g(x)\) and \(y=f(x)g(x)\) and hence sketch the graphs
• apply knowledge of graphical relationships to solve problems in practical and abstract contexts

Ref: https://educationstandards.nsw.edu.au/