Determine which domain corresponds to the function described below: The function represents the height of a child from birth to first grade.

Partly increasing and partly decreasing.

Which domain corresponds to the described function: The function represents the velocity of a stone after being dropped from a great height as a function of time.

Partly increasing and partly decreasing

Determine the increasing and decreasing domain of the following function: The function describes a student's grades throughout the year.

Partly increasing and partly decreasing.

Increasing and Decreasing Function Intervals Practice Problems

Understanding Increasing functions

Complete explanation with examples

Increasing functions

What is an Increasing Function?

An increasing function is a type of relationship where, as you move to the right on the graph (increasing the $x$ -value), the $y$ -value also gets bigger. It’s like climbing a hill—the higher you go (the more you increase $x$ ), the more your height (the $y$ -value) increases.

We will say that a function is increasing when, as the value of the independent variable $X$ increases, the value of the function $Y$ increases.

How to Spot an Increasing Function:

On a Graph: The line or curve goes upwards as you move from left to right.
In Numbers: For any two xxx-values, if the second number is larger than the first $x_2 > x_1$ , then the second $y$ -value will also be larger than the first $f(x_2) > f(x_1)$ .

Real-Life Example:

Think about saving money in a piggy bank. Every day you add more coins, and the total amount of money keeps going up. That’s an increasing function in action—your savings are the $y$ -values, and the number of days is the $x$ -values.

Fun Fact:

If the line or curve never stops going up, it's called strictly increasing. If it flattens for a bit before going up again, it's just increasing.

let's see an example of strictly increasing linear function:

Detailed explanation

Practice Increasing functions

Test your knowledge with 17 quizzes

Determine the domain of the following function:

A function describing the charging of a computer battery during use.

Examples with solutions for Increasing functions

Step-by-step solutions included

Exercise #1

Is the function shown in the graph below decreasing?

Step-by-Step Solution

The graph presented is a straight line. To determine whether the function is decreasing, we need to examine the slope of this line.

The line has a negative slope, as it moves downward from left to right. A function is considered decreasing when its slope is negative.

In formal terms, for a linear function expressed as $y = mx + c$ , if the slope $m$ is negative, the function is decreasing over its entire domain.

From the graph, it's evident that the line has a negative slope, thus indicating that the function is indeed decreasing.

Therefore, the answer to the problem is Yes.

Answer:

Yes

Exercise #2

Is the function in the graph decreasing?

Step-by-Step Solution

To analyze whether the function in the graph is decreasing, we must understand how the function's behavior is defined by its graph:

Step 1: Examine the graph. The graph presented is a horizontal line.
Step 2: Recognize the properties of a horizontal line. Horizontally aligned lines correspond to constant functions because the $y$ -value remains the same for all $x$ -values.
Step 3: Define the criteria for a function to be decreasing. A function decreases when, as $x$ increases, the value of $f(x)$ decreases.
Step 4: Apply this criterion to the horizontal line. Since the $y$ -value is constant and does not decrease as $x$ moves rightward, the function is not decreasing.

Therefore, the function represented by the graph is not decreasing.

Answer:

No

Exercise #3

Is the function in the graph below decreasing?

Step-by-Step Solution

To determine if the function is decreasing, we will analyze the graph visually:

The graph shows a line connecting from the bottom-left to the top-right of the graph area, indicating the line has a positive slope. This type of graph indicates the function is increasing, not decreasing.

A decreasing function means its value goes down as $x$ increases, which is equivalent to having a negative slope.

Since the graph appears with a positive slope, the function is not decreasing.

Thus, the correct choice to the problem, which asks if the function in the graph is decreasing, is No.

Answer:

No

Exercise #4

Does the function in the graph decrease throughout?

Step-by-Step Solution

To solve this problem, we'll begin by examining the graph of the function provided:

Step 1: Observe the graph from left to right along the x-axis.
Step 2: Look for any intervals where the function value (y-coordinate) does not decrease as the x-value increases.
Step 3: Pay special attention to segments where the graph might look horizontal or rising.

Upon inspecting the graph, we find:

- There are sections where the function's y-values appear to remain constant or potentially rise as the x-values increase. Specifically, even if the function decreases in major portions, any interval where it doesn't means the function cannot be classified as decreasing throughout.

Thus, the function does not strictly decrease on the entire interval shown. Therefore, the solution to the problem is No.

Answer:

No

Exercise #5

Is the function in the graph decreasing?

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Verify the graph's overall path direction
Step 2: Confirm if the y-values are decreasing as we proceed from the left side of the graph to the right side (increasing x-values).

Now, let's work through each step:

Step 1: By examining the graph, the red line starts at a higher point on the y-axis and moves downward to a lower point as it moves horizontally across the x-axis from left to right.

Step 2: Since for every point, the red line descends as it progresses from the leftmost point to the rightmost, this indicates a consistent decrease in the y-values.

Therefore, the solution to the problem is Yes, the function in the graph is decreasing.

Answer:

Yes

Frequently Asked Questions

Everything you need to know about Increasing functions

What is the difference between increasing and decreasing functions?

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An increasing function has y-values that get larger as x-values increase (moving left to right on a graph, the line goes up). A decreasing function has y-values that get smaller as x-values increase (the line goes down from left to right). Mathematically, a function is increasing when f(x₂) > f(x₁) whenever x₂ > x₁.

How do you find increasing intervals of a quadratic function?

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For quadratic functions, find the vertex using x = -b/(2a). If a > 0 (parabola opens up), the function decreases before the vertex and increases after. If a < 0 (parabola opens down), the function increases before the vertex and decreases after. The vertex marks the transition point between increasing and decreasing intervals.

What does strictly increasing mean in mathematics?

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A strictly increasing function continuously rises without any flat sections - it never stays constant or decreases. In contrast, a regular increasing function can have flat portions where it remains constant before continuing to increase. Strictly increasing functions have a steeper, more consistent upward trend.

How can you tell if a function is increasing from its graph?

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Look at the direction of the line or curve as you move from left to right: • If it slopes upward, the function is increasing • If it slopes downward, the function is decreasing • If it's horizontal, the function is constant • Check specific intervals, as functions can increase in some areas and decrease in others

What are some real-world examples of increasing functions?

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Common examples include: savings account balance over time (with regular deposits), height of a growing plant over months, temperature rising throughout a spring day, or distance traveled at constant speed over time. These all show one variable consistently increasing as another variable increases.

How do you solve y = -(x+3)² type problems for increasing intervals?

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Step-by-step process: 1. Expand: y = -x² - 6x - 9 2. Identify coefficients: a = -1, b = -6, c = -9 3. Find vertex: x = -(-6)/(2×-1) = -3 4. Since a < 0, parabola opens down (maximum at vertex) 5. Function increases for x < -3 and decreases for x > -3

Why is the vertex important for finding increasing and decreasing intervals?

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The vertex represents the turning point of a parabola - where the function changes from increasing to decreasing (or vice versa). For upward-opening parabolas (a > 0), the function decreases before the vertex and increases after. For downward-opening parabolas (a < 0), the function increases before the vertex and decreases after.

Can a function be both increasing and decreasing?

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Yes, many functions have different intervals where they increase and decrease. For example, a parabola y = x² decreases for x < 0 and increases for x > 0. When analyzing functions, we identify specific intervals or domains where the function exhibits increasing or decreasing behavior, rather than describing the entire function with one term.

Continue Your Math Journey

Master Increasing functions first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

Functions for Seventh Grade Increasing and Decreasing Intervals (Functions)Decreasing function Constant Function Decreasing Interval of a function Increasing Intervals of a function Domain of a Function Indefinite integral Inputing Values into a Function

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems

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Increasing and Decreasing Function Intervals Practice Problems

Understanding Increasing functions

Increasing functions

What is an Increasing Function?

How to Spot an Increasing Function:

Real-Life Example:

Fun Fact:

Practice Increasing functions

Examples with solutions for Increasing functions

Frequently Asked Questions

What is the difference between increasing and decreasing functions?

How do you find increasing intervals of a quadratic function?

What does strictly increasing mean in mathematics?

How can you tell if a function is increasing from its graph?

What are some real-world examples of increasing functions?

How do you solve y = -(x+3)² type problems for increasing intervals?

Why is the vertex important for finding increasing and decreasing intervals?

Can a function be both increasing and decreasing?

More Increasing functions Questions

Increasing and Decreasing Intervals of a Function

Continue Your Math Journey

Suggested Topics to Practice in Advance

Topics Learned in Later Sections

Practice by Question Type