Rate of Change in Functions and Direct Variation

Rate of Change in Functions

• Height of a falling object modeled by h(t) = 300 - 16t²; average rate of fall in the first 3 seconds calculated using (h(3) - h(0)) / 3.
• Average speed of a ball between 2 seconds and 8 seconds is 0.5 meters per second.
• If y varies directly with x, y is 180 when x is n, and y is n when x is 5; value of n is 30.

Average Rate of Change

• Average rate of change in f(x) over [2, 9] determined by (f(9) - f(2)) / (9 - 2).
• For direct variation, when y is 48 at x = 6, the equation used to find y at x = 2 is y = (48/6)(2).
• Cost of soup varies directly with cans; determined values signify relationships to corresponding graphs.

Ticket Sales and Direct Variation

• For T(d), the average rate of change being 0 implies the same number of tickets sold on the fourth and tenth days.
• Relation that includes the ordered pair (2, 7) in direct variation takes the form y = (7/2)x.
• Average rate of change of g(x) is given by (g(7) - g(4)) / (7 - 4) = 5/6.

Temperature Change and Volume of Sphere

• Maria's temperature function B(t) shows an average rate of change of 8 between t = 3 and t = 7, indicating a temperature increase of 32 degrees.
• Volume of a sphere expressed as V = (3/4)πr³ must be solved for r.

Graphs and Relations

• A graph representing y = 2x must hold true for all real numbers x.
• Evaluations of functions like f(x) = 10 - 2x can yield specific results such as f(7) = -4.
• For b(x) = |x + 4|, calculating b(-10) gives a value of 6.

Functions and Sets

• To represent the same relation as the ordered pair set {(-6,4),(-4,0),(-3,2),(-1,2)}, the correct table matches these coordinates.
• Using h(f) = -2(t + 5)² + 4, finding h(-8) results in -14.
• Identifying a one-to-one function from a list emphasizes unique outputs for every input.

Summary of Range and Relationships

• The range of graphed functions is crucial; example values include -3 for specific functions.
• Various relations like the sets of points and their respective graphs showcase direct correlations and changes in function outputs.