How is the IS curve derived?

There is no inconsistency here. Following Blanchard et al. Macroeconomics an European Perspective (which is by the way same textbook where you got that picture from) the goods market can be described as:

$$ Y = C(Y − T) + I(Y, i) + G quad (1)$$

where, $Y$ is the output, $C$ consumption, $T$ taxes, $I$ investment $i$ interest and $G$ gov. spending. Then if we assume that consumption follows:

$$C = c_0 + c_1(Y − T) quad (2)$$

and that investment function is given by:

$$I = bar{I} + d_1Y − d_2i quad (3) $$

then the goods market equilibrium will be given as:

$$ Y = frac{1}{1-c_1-d_1}A - frac{d_2}{1-c_1-d_1}i quad(4)$$

where $A=[c_0 + bar{I} + G − c_1T]$. Finally the IS curve is derived just by solving the above equation for $i$:

$$i = frac{1}{d_2}A - frac{1-c_1-d_1}{d_2} Y quad (5)$$

Now this last expression clearly corresponds to the IS from the graph. As the equation lay bare the relationship between interest rate and output is still the same - lower interest rate is associated with higher output. The intuition behind negative relationship between investment and the interest rate is determined by the fact that the higher the interest rate, the lower the incentive for a company to borrow to make new investment. Higher investment means also higher output $Y$ so the lower the interest rate the higher output will be.

Really the upper graph just plots the equation (4) although rearranged in different way and the bottom graph is the equation (5). So the relationship was already negative and same in both upper and lower graph. The graphs also might not be drawn to scale - they dont look to me like they are generated by specifying the underlying functions in lets say tikz picture in LaTex.

It's the other way around , as intrest rates decrease , planned investment increases so the output increase , however whole concept moves along real intrest rates , output doesn't solely contribute to change in interst rates , but change in intrest rates does directy contribute to output, Example- if real interst rate is 20% , expected return on investment is 19%, we choose not to opt such investment does output decreases , if expected return is 25%i opt for investment thus increasing output in whole economic perspective , assumption - inflation remains constant