verticle and horizontal lines
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If you mean: y -4 = 3(x-1) then y = 3x+1 which is a straight line equation
The graphs of y = 5x - 2 and y = x - 2 will have different slopes but with the same y intercepts.
This is finding how far the point is on the x-axis and how far the point is on the y-axis.
f(x) and g(x) are just names of generic functions - they could be anything. In any specific case, where they intersect depends on how the functions are defined. In general, to find out where they intersect you can solve for: f(x) = g(x) Replacing the corresponding expressions for each function of course.
If the two at the end of these are exponents, like x^2, then these graphs would be reflections across the x-axis. Their graphs would be two parabolas. f(x) pointing up, and g(x) pointing down.