Question 5.6: Starting with equation 5.6, show that the equations for the ...
Starting with equation 5.6, show that the equations for the slope, y-intercept, and x-intercept in Figure 5.7(a) are correct.
S_{spike} = k\left\lgroup C_A\frac{V_0}{V_\mathrm{f} }+C_s\frac{V_s}{V_\mathrm{f} } \right\rgroup (5.6)
We begin by rewriting equation 5.6 as
S_{spike} = \frac{kC_AV_0}{V_\mathrm{f}}+\frac{kC_s}{V_\mathrm{f}} V_s
which is in the form of the linear equation
Y = y-intercept + slope × X
where Y is S_{spike} and X is V_s. The slope of the line, therefore, is kC_S/V_\mathrm{f}, and the y-intercept is kC_AV_o/V_\mathrm{f}. The x-intercept is the value of X when Y is 0, or
0 = \frac{ k C_A V_0 }{ V_\mathrm{f} }+\frac{ k C_S}{ V_\mathrm{f} } \times (\text{x-intercept})
\text{x-intercept}=-\frac{( kC_A V_0 / V_\mathrm{f}) }{ k C_S / V_S } =-(\frac{ C_A V_0 }{ C_S })
Thus, the absolute value of the x-intercept is C_AV_o/C_S.