Section 3.5 - Linear And Nonlinear Relations

The formula P = 4s is linear because it represents the perimeter of a square. The perimeter is directly proportional to the length of one side of the square, so as the length of the side (s) increases, the perimeter (P) also increases by a factor of 4.

If the relationship is linear, it means that there is a constant rate of change between the values. This constant rate of change is reflected in the finite differences, which are calculated by subtracting consecutive values in the difference table. For every row in the difference table, the finite differences will be the same and constant, indicating the linear relationship between the values.

The equation y = xz represents a linear relation because it is a simple equation where y is directly proportional to x and z. The variables are only raised to the power of 1, and there are no other non-linear operations involved.

The equation y = 2/3x^2 + 11 represents a nonlinear relation because it contains a squared term (x^2). In a linear relation, the highest power of the variable should be 1.

The missing value in the linear relation can be determined by observing the pattern in the given values. From -5 to -8, there is a decrease of 3. Similarly, from -8 to -11, there is a decrease of 3. Therefore, continuing the pattern, the next decrease of 3 would result in the missing value of -14.

If a relation is linear, it means that the graph is a straight line. The highest exponent of its equation is 1, which indicates that the equation is in the form y = mx + b. The first differences are constant, meaning that the difference between consecutive y-values remains the same. However, a linear equation cannot have a simplified fraction with x in the denominator. This is because a fraction with x in the denominator would indicate a non-linear relationship, as it would involve division and not a constant ratio between x and y.

A table of values represents a weak, non-linear relationship if the data has no obvious pattern. This means that there is no clear trend or correlation between the variables being measured. The values may be scattered or random, indicating a lack of relationship or a very weak association between the variables.

If a relation is non-linear, it means that the graph is not a straight line. This implies that the highest exponent of its equation is not 1, as a linear equation would have an exponent of 1. Additionally, the first differences are not constant in a non-linear relation. However, it is not necessary for the graph to always pass through the origin in a non-linear relation. Therefore, the statement "The graph always passes through the origin" is not true for a non-linear relation.

Based on the table, Josephine's distance increases over time. Since the marathon distance is 42km, we need to find the time at which Josephine reaches this distance. The table shows that at 236.25 minutes, she has covered a distance of 41km. Since she is still short of the 42km mark at this time, we can conclude that it will take her additional time to reach 42km. The next time interval shown in the table is 240 minutes, at which Josephine's distance is 42km. Therefore, the answer is 240 minutes.