3.3 Rates of Change and Behavior of Graphs
3 min read•june 24, 2024
Rates of change reveal how functions evolve over time or space. They're crucial for understanding trends and predicting future values. This concept helps us grasp the behavior of graphs, showing where functions increase, decrease, or remain constant.
Graphical behavior analysis is key to interpreting data and solving real-world problems. By identifying local and absolute , we can pinpoint optimal solutions in various scenarios, from maximizing profits to minimizing costs in business applications.
Rates of Change
Average rate of change calculation
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- Measures how quickly a function changes over a specific interval calculated using the formula:
- and are the endpoints of the interval ( is the starting point, is the ending point)
- Represents the slope of the connecting the points and (the line that intersects the graph at the interval endpoints)
- Interpretation of
- Positive value indicates the function is increasing on average over the interval (moving upward from left to right)
- Negative value indicates the function is decreasing on average over the interval (moving downward from left to right)
- Zero value indicates the function remains constant on average over the interval (not changing in height)
Intervals of function behavior
-
- Function values consistently rise from left to right (going uphill)
- Slope of the is positive at every point in the interval (the line that touches the graph at one point)
-
- Function values consistently fall from left to right (going downhill)
- Slope of the tangent line is negative at every point in the interval
-
- Function values remain the same throughout the interval (flat line)
- Slope of the tangent line is zero at every point in the interval
Rates of Change and Differentiability
- : A function that describes the of the original function at any given point
- : The rate of change at a specific point, represented by the derivative
- : A property of functions where there are no breaks or jumps in the graph
- : A function is differentiable if it has a derivative at every point, which requires both continuity and a well-defined tangent line at each point
Behavior of Graphs
Local extrema on graphs
- (peak, )
- Point where the function value is greater than or equal to the values in its immediate vicinity (the surrounding points)
- Occurs when the function changes from increasing to decreasing (going uphill then downhill)
- (valley, )
- Point where the function value is less than or equal to the values in its immediate vicinity
- Occurs when the function changes from decreasing to increasing (going downhill then uphill)
- Significance of local maxima and minima
- Identify the highest and lowest points within a specific region of the function (hills and valleys on the graph)
- Indicate where the function changes direction from increasing to decreasing or vice versa (turning points)
- Used in problems to find the best or worst-case scenarios within a given context (maximizing profit, minimizing cost)
Absolute extrema from graphs
- ()
- Highest function value over the entire (the tallest peak on the graph)
- No other point on the graph has a greater (nothing is higher)
- ()
- Lowest function value over the entire (the deepest valley on the graph)
- No other point on the graph has a smaller y-value (nothing is lower)
- Finding absolute extrema on a graph
- Identify the highest and lowest points on the graph visually (peaks and valleys)
- Consider the function values at the endpoints of the domain, if applicable (the far left and right points)
- Compare the y-values of local maxima, local minima, and endpoints to determine the absolute extrema (the highest high and lowest low)
Key Terms to Review (33)
Absolute Maximum: The absolute maximum of a function is the largest value the function attains over its entire domain. It represents the global maximum point, which is the highest point on the graph of the function, regardless of the local behavior of the function.
Absolute minimum: The absolute minimum of a function is the lowest point over its entire domain. It represents the smallest value that the function attains.
Absolute Minimum: The absolute minimum is the lowest possible value that a function can attain within a given domain. It represents the global minimum point on the graph of a function, which is the point where the function reaches its lowest value across the entire domain.
Average Rate of Change: The average rate of change measures the average change in the dependent variable over a given interval of the independent variable. It provides a summary of the overall rate of change between two points on a graph or in a function.
Constant Intervals: Constant intervals refer to the consistent change or difference between successive values in a sequence or function. This concept is particularly relevant in the context of rates of change and the behavior of graphs, as it helps analyze the patterns and trends within a given data set.
Continuity: Continuity is a fundamental concept in mathematics that describes the smooth and uninterrupted behavior of a function or graph. It is a crucial property that ensures a function's values change gradually without any abrupt jumps or breaks.
Decreasing function: A decreasing function is one where the value of the function decreases as the input increases. For any two points $x_1$ and $x_2$ where $x_1 < x_2$, $f(x_1) \geq f(x_2)$.
Decreasing Intervals: Decreasing intervals refer to the portions of a graph where the function's value decreases as the independent variable increases. This concept is closely tied to the rates of change and the overall behavior of a graph.
Derivative: The derivative is a fundamental concept in calculus that measures the rate of change of a function at a particular point. It represents the slope of the tangent line to the function's graph at that point, providing information about the function's behavior and how it is changing.
Differentiability: Differentiability is a fundamental concept in calculus that describes the smoothness and continuity of a function. It is closely related to the idea of the derivative, which measures the rate of change of a function at a particular point.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Electrostatic force: Electrostatic force is the force of attraction or repulsion between electrically charged objects. It follows Coulomb's Law, which states that the force is inversely proportional to the square of the distance between charges and directly proportional to the product of the charges.
Endpoint: An endpoint is a point at one end of a segment or the starting point of a ray. It represents the limit or boundary of a function on a given interval.
Extrema: Extrema, in the context of mathematics, refers to the maximum and minimum values that a function can attain within a given domain. This concept is crucial in understanding the behavior and properties of graphs, as well as the rates of change associated with functions.
Global maximum: A global maximum is the highest point over the entire domain of a function. For polynomial functions, it is where the function attains its greatest value.
Global Maximum: The global maximum of a function is the highest point or value that the function attains over its entire domain. It represents the absolute maximum value of the function, as opposed to a local maximum which is the highest point within a specific region of the function's graph.
Global minimum: The global minimum of a function is the lowest point over its entire domain. It represents the smallest value that the function can take.
Global Minimum: The global minimum of a function is the point on the function's graph where the function achieves its absolute lowest value. It represents the point at which the function reaches its global or overall minimum, as opposed to a local minimum which is the lowest point in a particular region of the graph.
Increasing Intervals: Increasing intervals refer to the periods within a function's graph where the function is consistently rising or growing larger. This concept is closely tied to the rate of change and the overall behavior of the function's graph.
Instantaneous Rate of Change: The instantaneous rate of change refers to the rate of change of a function at a specific point in time or at a particular value of the independent variable. It represents the slope or steepness of the tangent line to the graph of the function at that point, and it measures how quickly the function is changing at that instant.
Local extrema: Local extrema are points on a graph where a function reaches a local maximum or minimum value. These points represent the highest or lowest values within a specific interval of the function.
Local maximum: A local maximum of a function is a point at which the function's value is higher than that of any nearby points. It is not necessarily the highest point on the entire graph, but rather within a specific interval.
Local Maximum: A local maximum is a point on a graph where the function value is greater than or equal to the function values at all nearby points. It represents a relative high point on the graph, even if it is not the absolute highest point.
Local minimum: A local minimum is a point on the graph of a function where the function value is lower than all nearby points. It represents the lowest value within a specific interval.
Local Minimum: A local minimum is a point on a graph where the function value is less than or equal to the function values at all nearby points. It represents a point where the function reaches a minimum within a certain region, but not necessarily the overall minimum of the function.
Optimization: Optimization is the process of finding the best or most favorable solution to a problem or situation, typically by maximizing desired outcomes and minimizing undesirable ones. It involves selecting the optimal values of variables or parameters to achieve the most favorable outcome under given constraints.
Rate of change: Rate of change measures how one quantity changes in relation to another. In algebra, it often refers to the slope of a line, indicating how the dependent variable changes with respect to the independent variable.
Relative Maximum: A relative maximum is a point on a graph where the function value is greater than or equal to the function values in the immediate surrounding region. It represents a local peak or high point on the graph of a function.
Relative Minimum: A relative minimum is a point on a graph where the function value is less than or equal to the function values at all nearby points. It represents a local low point on the graph, where the function temporarily decreases before increasing again.
Secant Line: A secant line is a straight line that intersects a curve at two distinct points. It provides a way to measure the average rate of change of a function between two points on the curve, which is an important concept in understanding the behavior of graphs and analyzing rates of change.
Tangent Line: A tangent line is a straight line that touches a curve at a single point, without crossing or intersecting the curve. It represents the instantaneous rate of change of the function at that specific point.
Y-value: The y-value, also known as the ordinate, refers to the vertical coordinate of a point on a graph. It represents the value or measurement along the y-axis, which is typically the vertical axis in a two-dimensional coordinate system.