Final answer:
The tools that help in determining if a function is increasing include a graphing calculator and a derivative calculator. Graphing calculators allow for visual assessment, while derivative calculators provide information on the slope. The correct option indicating a function increasing at x = 3 with a positive slope that decreases in magnitude is a quadratic function like y = x².
Step-by-step explanation:
To determine where a function is increasing, several tools can be used. A function is said to be increasing on an interval if the output values increase as the input values increase. This is often visually represented on a graph where the function shows an upward trend from left to right. The following options can help you determine if a function is increasing:
- In a graphing calculator, you can plot the function and visually inspect if it is increasing.
- Using a derivative calculator can show whether the slope of the tangent to the function is positive (indicating an increase).
- By evaluating the slope at various points, you can determine if the function is consistently moving upwards.
- An inequality solver can tell you the range of values for which a function is above a certain threshold, but this does not specifically indicate increasing behavior.
The TI-83, 83+, or 84 calculators also provide ways to assess if a function is increasing. If we consider the graph of a function provided as an example, such as y = a + bx, if b > 0, the line will slope upward to the right, indicating an increasing function. Analyzing the slope and its behavior as the input values change provides a method to determine if the function is increasing or not.
Considering the question provided, at x = 3, if a function f(x) has a positive value with a positive slope that is decreasing in magnitude with increasing x, the function could still be increasing but at a decreasing rate. So, the correct function representing these characteristics could be b. y = x², as it is a quadratic function that increases with a decreasing rate as x gets larger.
Therefore, the graphing calculator and the derivative calculator are both good tools for determining if a function is increasing.