Increasing and Decreasing Functions Practice Problems

Master function intervals with step-by-step practice problems. Learn to identify increasing and decreasing behavior from graphs and equations.

📚Practice Identifying Function Behavior and Intervals
  • Determine decreasing intervals from quadratic function graphs and equations
  • Identify where functions are positive or negative using x-intercepts
  • Find vertex coordinates to locate maximum and minimum points
  • Analyze parabola direction using coefficient signs in standard form
  • Apply decreasing function definition: when x₂ > x₁, then f(x₂) < f(x₁)
  • Solve real-world problems involving increasing and decreasing relationships

Understanding Decreasing function

Complete explanation with examples

Decreasing function

What is a Decreasing Function?

A decreasing function is a type of relationship where, as you move to the right on the graph (increasing the xxx-value), the yy-value gets smaller. It’s like going downhill—the farther you go (the more you increase xx), the lower your height (the yy-value) becomes.

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

How to Spot a Decreasing Function:

  1. On a Graph: The line or curve goes downward as you move from left to right.
  2. In Numbers: For any two xx-values, if the second number is larger than the first \(x_2 > x_1\​), then the second yy-value will be smaller than the first f(x2)<f(x1)f(x_2) < f(x_1).

Real-Life Example:

Think about eating a stack of cookies. Every time you eat one, the number of cookies left in the stack gets smaller. That’s a decreasing function—your yy-value (cookies left) decreases as your xx-value (number of cookies eaten) increases.

Fun Fact:

If the line or curve always goes down without stopping, it's called strictly decreasing. If it flattens for a bit before going down again, it’s just decreasing.

Let's see an example of strictly decreasing linear function on a graph:

Decreasing function

Detailed explanation

Practice Decreasing function

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 Decreasing function

Step-by-step solutions included
Exercise #1

Is the function shown in the graph below decreasing?

yx

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 y = mx + c , if the slope m 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?

yx

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 y -value remains the same for all x x -values.
  • Step 3: Define the criteria for a function to be decreasing. A function decreases when, as x x increases, the value of f(x) f(x) decreases.
  • Step 4: Apply this criterion to the horizontal line. Since the y y -value is constant and does not decrease as x 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?

yx

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 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?

YYYXXX

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? yx

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 Decreasing function

How do you find the decreasing interval of a quadratic function?

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For a quadratic function f(x) = ax² + bx + c, first find the vertex x-coordinate using x = -b/(2a). If a > 0 (parabola opens up), the function decreases for x < vertex x-value. If a < 0 (parabola opens down), the function decreases for x > vertex x-value.

What makes a function decreasing vs strictly decreasing?

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A function is decreasing if f(x₂) ≤ f(x₁) whenever x₂ > x₁. A function is strictly decreasing if f(x₂) < f(x₁) whenever x₂ > x₁. Strictly decreasing means the function never stays flat - it always goes down as you move right.

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

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Look at the graph from left to right. If the line or curve goes downward (like going downhill), the function is decreasing in that interval. The y-values get smaller as x-values increase.

What's the difference between where a function is decreasing and where it's negative?

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A function is decreasing when its values get smaller as x increases (slope behavior). A function is negative when its y-values are below the x-axis (sign of output). These are completely different concepts - a function can be decreasing but positive, or increasing but negative.

How do you find where a quadratic function changes from increasing to decreasing?

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The turning point occurs at the vertex. For f(x) = ax² + bx + c, calculate x = -b/(2a) to find the vertex x-coordinate. This is where the function switches from increasing to decreasing (or vice versa).

Can a linear function be both increasing and decreasing?

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No, a non-horizontal linear function is either always increasing or always decreasing throughout its entire domain. Only horizontal lines (constant functions) are neither increasing nor decreasing.

What does it mean when a function has intervals of increase and decrease?

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This means the function changes behavior at different x-values. For example, a parabola might decrease until it reaches its vertex, then increase afterward. The function has different behaviors in different intervals of its domain.

How do you write interval notation for decreasing functions?

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Use parentheses for open intervals and brackets for closed intervals. For example, if a function decreases for x < -1, write (-∞, -1). If it decreases for x ≤ 2, write (-∞, 2]. Always use parentheses with infinity symbols.

More Decreasing function Questions

Click on any question to see the complete solution with step-by-step explanations

Increasing and Decreasing Intervals of a Function

Finding Decreasing Intervals: Analyzing Function Behavior from Graph Identifying Decreasing Domains in a Piecewise Function Graph Finding Decreasing Intervals on a Curved Function Graph: [0,6] Analysis Finding Function Increase Domain with Vertical Line x=-0.8: Graph Analysis Finding Decreasing Intervals: Graph Analysis with x=0.65 Reference Line

Continue Your Math Journey

Master Decreasing function first, then advance to these related topics that build on your fraction skills

Suggested Topics to Practice in Advance

Ways to Represent a FunctionRepresenting a Function Verbally and with TablesGraphical Representation of a FunctionAlgebraic Representation of a FunctionNotation of a FunctionRate of Change of a FunctionVariation of a FunctionRate of change represented with steps in the graph of the functionRate of change of a function represented graphicallyConstant Rate of ChangeVariable Rate of ChangeRate of Change of a Function Represented by a Table of Values

Topics Learned in Later Sections

Functions for Seventh GradeIncreasing and Decreasing Intervals (Functions)Increasing functionsConstant FunctionDecreasing Interval of a functionIncreasing Intervals of a functionDomain of a FunctionIndefinite integralInputing Values into a Function

Practice by Question Type

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