Apologies in advance if the question is wordy.
Let's say that I have $75\text{%}$ in a class.The grade is calculated with category weights. Here are the categories and their weights:
Homework: $0.1 (10\text{%})$
Projects: $0.2 (20\text{%})$
Exams: $0.7 (70\text{%})$
Here are the assignments with their scores, percentages, and categories.
Homework Assignment: $10/10 (100\text{%})$
Project: $8/10 (80\text{%})$
Exam: $7/10 (70\text{%})$
The grade for this class is $70\text{%}$. Now, what if I wanted to get it up to $80\text{%}$? I know that if I get $9/10$ on a test or $90\text{%}$, the grade goes up to $82\text{%}$. How could I represent this using a mathematical formula or function?
Insight on how the grades are calculated:
Each category has a certain number of points possible and points earned. Each time that a new assignment is added, the category's points possible go up by the denominator of the new assignment's score, and the category's points earned go up by the numerator of the new assignment's score.
I'm not sure I understand exactly what you're trying to do. Let $g$ be the course grade, $h$ the homework score, $p$ the project score, and $t$ the test score. We are given $$g = .1h+.2p+.7t$$
As you say, if $h=1, p=.8,t=.7$ then $$g=.10+.16+.49=.75$$ If you get $90\%$ on the next exam, raising your exam average to $.8$ then your grade becomes $$g=.10+.16+.56=.82$$
Are you trying to solve $$.8=.10+.16+.7t?$$ This gives $$t=\frac{54}{70}\approx.7714$$ so to get a grade of at least $80\%$ you need to raise your test average to $78\%$ and assuming that the second exam has the same number of points as the first, you need to score $86\%$, because $$\frac{70+86}2 = 78$$
If this doesn't exactly answer your question, I hope it at least shows how you can answer it yourself.