How do I mathematically connect these performance chart graphs that have different algebraic formulas?

Question

I have 5 graphs, each representing a different factor, where the answer will be found on the right side (y axis) of the final graph. I have an image I could send but don’t see how to attach here.

The way it should work is starting at the left graph I find where my given x,y is then I move straight to the right until hitting the y axis of graph #2. From here I follow the slope of the graphed line until reaching my next given X factor. At this point I move directly right again and the process repeats. The issue I am having is that I want to be able to do this mathematically to get the exact answer versus eyeballing the charts. Again, I have the individual formulas for each line but I can’t figure out how to mathematically “follow the slope until hitting a given x”. Thank you for any help you can offer!

Again I can send a picture through email or however if it would help.

Answer 1

Totally get what you’re trying to do you’re “walking” across a row of charts, letting the height you land at on one chart become the starting height on the next. You can do that exactly with a simple composition formula.

Core idea (linear lines)

Suppose chart i uses a line with slope mi​. You “enter” that chart at x=x0,i (often the y-axis of that chart) at height yi−1. The unique line with slope mi that passes through (x0,i, yi−1) is

y = yi −1+ mi (x−x0,i).

You need the value at your given x=Xi for that chart, so

 yi = yi−1+mi (Xi−x0,i) .

Start with your known (X1, y1) on chart 1. If you must move from X1​ over to the next chart’s entry x0,2 along chart 1’s slope m1​, first compute the exit height:

yexit,1=y1+m1 (x0,2−X1).

then feed that into chart 2:

y2=yexit,1+m2 (X2−x0,2), and repeat across all charts. It’s just chaining the same anchor-and-slope step.

If your chart gives a full line y = aix + bi

Anchor it so you start at yi−1y​ when x=x0,i. Solve yi−1=aix0,i+bi⇒bi=yi−1−aix0,i. Then

yi = aiXi+(yi−1−aix0,i) = yi−1+ai (Xi−x0,i),which is the same formula with mi=ai​.

Tiny example:

• Chart 1 slope m1=0.4m1=0.4m1​=0.4. You’re at (X1=3, y1=10). Next chart’s entry is x0,2=0.

Exit height: yexit,1=10+0.4 (0−3)=8.8

• Chart 2 slope m2 = −1.2, target X2=5, entry x0,2=0.

y2=8.8+(−1.2) (5−0) = 8.8− 6 = 2.8.

Keep chaining that step for all five charts and you’ll get the exact final y without eyeballing.

If one of your charts isn’t linear, use the same anchoring trick: modify its formula so it passes through (x0,i, yi−1), then evaluate at Xi. If you can paste the formulas (or a photo), I can write the explicit composed expression for your whole pipeline.