ClassX Video Lessons Subjects Algebra Introduction to increasing, decreasing, positive or negative intervals | Algebra I
In this article, we’re going to dive into how a function behaves by looking at its graph, represented as ( y = f(x) ). We’ll figure out when the function is positive or negative and when it’s increasing or decreasing.
A function is positive when its values are above zero, meaning the graph is above the x-axis. Here’s how we can find these intervals:
So, we can say the function is positive when:
On the flip side, a function is negative when its values are below zero, meaning the graph is below the x-axis. Here’s how to spot these intervals:
Therefore, the function is negative when:
A function is increasing when the value of ( y ) goes up as ( x ) increases. You can tell by looking at the slope of the graph. Here’s when the function is increasing:
At point ( d ), the function changes from increasing to decreasing, so we don’t include ( d ) in the increasing interval.
Thus, the function is increasing when:
A function is decreasing when the value of ( y ) goes down as ( x ) increases. Here’s when the function is decreasing:
At points ( d ) and ( e ), the function is neither increasing nor decreasing.
Thus, the function is decreasing when:
In summary, we’ve explored how to determine when a function is positive or negative and when it’s increasing or decreasing. It’s important to note that these intervals don’t always match up. Understanding these differences is key to fully analyzing how a function behaves.
Using graphing software or a graphing calculator, plot different functions and identify the positive and negative intervals. Mark these intervals on the graph and explain why the function is positive or negative in those regions. This will help you visualize the concepts discussed in the article.
Work in pairs to create a set of functions. Exchange your functions with another pair and identify the increasing and decreasing intervals. Discuss your findings with your classmates to ensure everyone understands how to determine these intervals.
Create a storyboard that illustrates the behavior of a function over different intervals. Use drawings or digital tools to show when the function is positive, negative, increasing, or decreasing. Present your storyboard to the class to demonstrate your understanding.
Research a real-world scenario where understanding the behavior of a function is crucial, such as economics or physics. Describe how identifying positive/negative and increasing/decreasing intervals can provide insights into the situation. Share your findings in a short presentation.
Create a quiz for your classmates that includes questions about positive and negative intervals, as well as increasing and decreasing behavior of functions. Use multiple-choice and short-answer questions to test their understanding. Exchange quizzes with a partner and discuss the answers.
Function – A relation between a set of inputs and a set of permissible outputs, where each input is related to exactly one output. – The function f(x) = 2x + 3 represents a straight line when graphed on a coordinate plane.
Positive – Greater than zero; often used to describe numbers or values that are above zero on a number line. – The positive slope of the line indicates that the function is increasing.
Negative – Less than zero; often used to describe numbers or values that are below zero on a number line. – The negative value of the slope means the line is decreasing as it moves from left to right.
Interval – A range of numbers between two specified limits, often used to describe a portion of a graph or function. – The function is increasing over the interval from x = 1 to x = 5.
Increasing – Describes a function or sequence that rises as the input or index increases. – The graph shows that the function is increasing for all x greater than 2.
Decreasing – Describes a function or sequence that falls as the input or index increases. – The function is decreasing between the interval x = -3 and x = 0.
Graph – A visual representation of data or a function, typically using a coordinate system. – The graph of the quadratic function is a parabola opening upwards.
Values – The numerical quantities that a function can take as inputs or outputs. – The values of the function f(x) = x^2 are always non-negative.
Slope – A measure of the steepness or incline of a line, often represented as the ratio of the rise over the run between two points on the line. – The slope of the line y = 3x + 2 is 3, indicating a steep incline.
Behavior – The manner in which a function acts or changes, especially as the input values become very large or very small. – The end behavior of the polynomial function shows that as x approaches infinity, the function value also approaches infinity.
