Increasing and Decreasing Intervals of a Function: Determine whether the graph is increasing or decreasing

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.

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.

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.

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.

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.

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

• Step 1: Visually inspect the graph to see if it is consistently sloping downward.
• Step 2: Apply the definition of a decreasing function.

Now, let's work through each step:
Step 1: Observing the graph, the function's graph is a line moving from the top left to the bottom right. This indicates it slopes downward as we move from left to right across the $x$-axis.
Step 2: According to the definition of a decreasing function, for any $x_1 < x_2$, it must hold true that $f(x_1) > f(x_2)$. Since the graph shows a line moving downward, this condition is satisfied throughout its domain.

Therefore, the function represented by the graph is indeed decreasing.

The final answer is Yes.

To determine whether the function $y = 4x - 2$ is increasing or decreasing, we follow these steps:

• Step 1: Identify the type of function we have. The given function is in the form of $y = mx + b$, which is a linear function.

• Step 2: Analyze the coefficient of $x$, known as the slope $m$. In our function, the slope $m$ is 4.

• Step 3: Understand the relationship between the slope and the rate of change. For linear functions, if the slope $m$ is positive, the function is increasing.

Since the slope $m = 4$ is positive, it means that as $x$ increases, $y$ also increases. Consequently, the function is increasing over its entire domain.

Therefore, the function $y = 4x - 2$ is Increasing.

To determine if the function $y = -2x$ is increasing or decreasing, we need to consider its slope.

The function $y = -2x$ is a linear function of the form $y = mx + b$, where:

• $m = -2$ (the slope)
• $b = 0$ (the y-intercept)

The slope $m$ is the rate of change of the function. For linear functions:

• If the slope $m > 0$, the function is increasing.
• If the slope $m < 0$, the function is decreasing.

In this case, the slope $m = -2$ is negative ($m < 0$). This indicates that as $x$ increases, $y$ decreases, meaning the function is decreasing.

Therefore, the function $y = -2x$ is decreasing.

To determine if the function $y = -3x + 3$ is increasing or decreasing, we need to examine the slope of the function.

The function is in the form $y = mx + b$, where $m$ is the slope.

In this function, the slope $m = -3$.

The sign of the slope $m$ tells us whether the function is increasing or decreasing:

• If $m > 0$, the function is increasing.
• If $m < 0$, the function is decreasing.

Since $m = -3$ and it is less than zero, the function is decreasing.

Therefore, the function is decreasing.

To determine if the function $y = x - 1$ is increasing or decreasing, we will analyze its slope:

• Step 1: Identify the function as a linear function in the form $y = mx + b$ where $m = 1$ and $b = -1$.

• Step 2: Recall that for a linear function, if the slope $m > 0$, the function is increasing. Conversely, if $m < 0$, it is decreasing.

• Step 3: Calculate the slope: $m = 1$. Since $m = 1$ is positive, this means the function is increasing.

The behavior of the function depends on the sign of the slope. Here, because the slope is positive, the function $y = x - 1$ increases as $x$ increases across its entire domain.

Therefore, the function is Increasing.

To determine if the function $y = -x - 2$ is increasing or decreasing, we analyze its slope.

• Step 1: Identify the slope. The given function is in the form $y = mx + b$, where $m$ is the slope. Here, $m = -1$.
• Step 2: Analyze the slope. Since $m = -1$, which is less than 0, the slope is negative.
• Step 3: Conclude based on the slope. For a linear function, if the slope is negative ($m < 0$), the function is said to be decreasing.

Therefore, the function is Decreasing.

To determine whether the function $y = 2x - 3$ is increasing or decreasing, we need to analyze the slope of the linear function.

We start by identifying the equation given: $y = 2x - 3$.

This equation is in the standard linear form, $y = mx + b$, where:
$m$ is the slope, and
$b$ is the y-intercept.

The slope $m$ in this equation is $2$. In the context of linear functions:

• If $m > 0$, the function is increasing.

• If $m < 0$, the function is decreasing.

• If $m = 0$, the function is constant (neither increasing nor decreasing).

In our function, the slope $m = 2$, which is greater than zero. Therefore, we can conclude that the function $y = 2x - 3$ is increasing.

As a result, the function is increasing.

To determine whether the function $y = 3 - x$ is increasing or decreasing, we need to examine its slope.

The given function is in the standard linear form $y = mx + b$, where $m$ is the slope. For the function $y = 3 - x$, the slope $m$ is $-1$.

The behavior of a linear function is determined by its slope:

• If the slope $m > 0$, the function is increasing.
• If the slope $m < 0$, the function is decreasing.

In this case, since the slope $m = -1$, which is less than zero, the function is decreasing.

Thus, the function $y = 3 - x$ is decreasing.

The correct choice is choice 2: Decreasing.

To solve this problem, we need to determine whether the linear function $y = 2x + 2$ is increasing or decreasing.

Linear functions are represented by the equation $y = mx + b$, where $m$ is the slope. The slope indicates the rate at which the function increases or decreases as we move along the x-axis.

For the given function $y = 2x + 2$, the slope $m$ is 2. This is a positive number.

A positive slope indicates that as $x$ increases, $y$ also increases. Therefore, the function is increasing.

Since a positive slope in the linear function suggests an increasing nature, we can conclude that the function is growing.

Therefore, the function $y = 2x + 2$ is Increasing.

To determine whether the function $y = 3x - 1$ is increasing or decreasing, we need to examine its slope. The function is in the form $y = mx + b$, where $m$ is the slope.

For the given function, the slope $m = 3$.

• If $m > 0$, the function is increasing.
• If $m < 0$, the function is decreasing.
• If $m = 0$, the function is constant (neither increasing nor decreasing).

Since the slope $m = 3$ is positive, the function is increasing.

Therefore, the function $y = 3x - 1$ is increasing.