Final answer:
Direct variation is a relationship where two variables change proportionally, represented by the equation y = kx. True statements include that their ratio is constant and the graph is a straight line through the origin. False statements commonly misunderstood include the direction of change and the value of k, which is never zero.
Step-by-step explanation:
A direct variation is a relationship between two variables in which one is a constant multiple of the other. This means that when one variable changes, the other changes in a proportional way. The equation to represent direct variation is y = kx, where k is the proportionality constant, indicating that y changes by k times when x changes.
Here are two true and two false statements about direct variations:
- True: If two variables are directly proportional, then their ratio is constant.
- True: A graph representing a direct variation is a straight line that passes through the origin (0, 0).
- False: A direct variation implies that an increase in one variable will always cause a decrease in the other. The correct statement: A direct variation implies that an increase in one variable will cause a proportional increase in the other.
- False: In a direct variation, the value of k can be zero. The correct statement: In a direct variation, the value of k cannot be zero since it is the ratio of the dependent variable to the independent variable, and division by zero is undefined.
Looking at examples of dependent and independent variables, the independent variable is the one that is changed or controlled to see its effect on the dependent variable, which is the variable being tested and measured.
The slope in a direct variation is the proportionality constant k, and it represents the rate of change between the dependent and independent variables. The y-intercept in a direct variation is always zero since the line passes through the origin.