Linear Function y=mx+b - Examples, Exercises and Solutions

For this problem, we need to determine the nature of the slope for a given straight line on a graph.

Based on the graph provided, the red line starts at a higher point on the left (Y-axis) and moves downward toward a lower point on the right (X-axis). This indicates that as one moves from left to right across the graph, the function decreases in value. Consequently, this is typical of a line that has a negative slope.

The slope of a line is typically defined as the "rise over the run," or the ratio of the change in the vertical direction to the change in the horizontal direction. Here, as we proceed from left to right, the line goes "downwards" (negative rise), establishing a negative slope.

Thus, we can conclude that the slope of the line is negative.

Therefore, the solution to the problem is Negative slope.

To solve this problem, follow these steps:

• Step 1: Observe the given graph and the plotted line.
• Step 2: Determine the direction of the line as it moves from left to right across the graph.
• Step 3: Understand that a line moving downwards from left to right represents a negative slope.

Now, let's work through these steps:

Step 1: The graph shows a straight line that starts higher on the left side and descends towards the right side.

Step 2: As the line moves from left to right, it descends. This is a key indicator of the slope type.

Step 3: A line that moves downward from the left side to the right side of the graph (decreasing in height as it proceeds to the right) is characteristic of a negative slope. Conversely, a positive slope would show a line ascending as it moves rightward.

Therefore, the solution to the problem is the line has a negative slope.

To determine the slope of the line, we'll examine the direction of the line segment on the graph:

• The line depicted moves from the top left, passing through a point with higher $y$-coordinate values, to the bottom right, ending at a point with lower $y$-coordinate values.
• This movement indicates that as $x$ increases (the direction to the right along the $x$-axis), the $y$-coordinate decreases.
• When the $y$-value reduces as the $x$-value grows, the slope $m$ is negative.

Since the line descends from left to right, the slope of the line is negative.

Therefore, the slope of the function is a negative slope.

To determine the slope of the line segment shown in the graph, follow these steps:

• Identify the line segment on the graph; it's shown as a red line from one point to another.
• Examine the direction the line segment travels from the leftmost point to the rightmost point.
• Visually analyze whether the line segment is rising or falling as it moves from left to right.

Here is the detailed analysis:
- The red line segment starts lower on the left side and ends higher on the right side.
- This suggests that as we move from left to right, the line is rising.
- In terms of slope, a line that rises as it moves from left to right has a positive slope.

Therefore, the slope of the line segment is positive.

Thus, the correct answer is Positive slope.

To solve this problem, let's evaluate the graph of the line provided:

• The line is visually represented as starting from the bottom left to the top right, moving upwards.
• In a standard Cartesian graph, a line that ascends as it progresses from left to right implies a positive change in the y-coordinate as the x-coordinate increases.
• This upward trajectory indicates that the slope, $m$, is positive.

Thus, the slope of the function is positive.

Therefore, the answer is Positive slope.

To solve this problem, let's analyze the given graph of the function to determine the slope's sign.

The slope of a line on a graph indicates the line's direction. A line with a positive slope rises as it moves from left to right, indicating that for every step taken to the right (along the x-axis), we move upward. Conversely, a line with a negative slope falls as it moves from left to right, meaning each step to the right results in moving downward.

Examining the graph provided, the red line starts higher on the left and goes downward towards the right visually. This indicates that the line is rising as it goes from left to right, which confirms it has a positive slope.

Therefore, the solution to the problem, regarding the slope of the line, is that it is a Positive slope.

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

• Step 1: Visual Inspection – Examine the red line on the graph to determine direction.
• Step 2: Determine Slope Direction – Ascertain if the line rises or falls as it moves from left to right.
• Step 3: Compare with Possible Answers – Verify which choice aligns with the determined slope direction.

Now, let's work through each step:
Step 1: The graph shows a red line segment, oriented in a manner that moves from left (lower) to right (higher).
Step 2: As the red line moves from the left toward the right side of the graph, it rises, indicating an upward trend and suggesting a positive slope.
Step 3: Given that the line increases from left to right, the slope is positive.

Therefore, the solution to the problem is Positive slope.

To determine the slope of the line shown on the graph, we perform a visual analysis:

• We examine the orientation of the line from left to right.
• The red line starts at a higher point on the left and descends to a lower point on the right.
• This indicates a downward movement, which corresponds to a negative slope.

Therefore, by observing the direction of the line, we conclude that the slope of the function is negative. This positional evaluation confirms that the correct answer is negative slope.

To solve this problem, we need to determine the slope of the line depicted on the graph.

First, understand that the slope of a line on a coordinate plane indicates how steep the line is and the direction it is heading. Specifically:

• A positive slope means the line rises as it goes from left to right.
• A negative slope means the line falls as it goes from left to right.

Let's examine the graph given:

• We see that the line starts at a higher point on the left and descends to a lower point on the right side.
• As we move from the left side of the graph towards the right, the line goes downwards.

This downward trajectory clearly indicates a negative slope because the line is declining as we move horizontally left to right.

Therefore, the slope of this function is Negative.

The correct answer is, therefore, Negative slope.

The given linear function is $y = 1 - 4x$. This can be rearranged to fit the standard format of a linear equation, $y = mx + b$, as $y = -4x + 1$.

In this form, the slope $m$ is simply the coefficient of $x$. Here, $m = -4$.

The slope of the linear function represents the rate of change of the function with respect to $x$, meaning for every unit increase in $x$, the value of $y$ decreases by 4 units.

Therefore, the rate of change of the function is $m = -4$, which is option 4 among the given choices.

Thus, the solution to the problem is $m = -4$.

The problem asks to find the rate of change of the linear function $y = x - 4$. This function is in the form of $y = mx + b$, where:

• $m$ is the slope, representing the rate of change of the function.
• $b$ is the y-intercept, but it is not relevant for finding the rate of change.

For the function $y = x - 4$, we can compare it with the standard form $y = mx + b$ to identify:

$m = 1$.

Therefore, the rate of change of the function is determined by the coefficient of $x$, which is 1.

Hence, the rate of change of the function is $m = 1$.

The correct answer is: $m = 1$.

To determine the characteristic of the function $y = 2 - 3x$, we will evaluate the slope:

• The given function is in the form $y = mx + b$, which indicates a linear equation. Here, $y = 2 - 3x$ can be rearranged as $y = -3x + 2$, showing that $m = -3$.
• The slope $m$ is $-3$.
• In a linear function, the sign of the slope $m$ determines the function's behavior:
• If the slope $m$ is positive ($m > 0$), the function is increasing.
• If the slope $m$ is negative ($m < 0$), the function is decreasing.
• If the slope $m = 0$, the function is constant.
• Since $m = -3$, which is negative, we conclude that the function is decreasing.

Therefore, the function described by $y = 2 - 3x$ is decreasing.

To solve this problem, we'll proceed with the following steps:

• Step 1: Find the slope $m$.
• Step 2: Use the slope and a point to find the $y$-intercept $b$.
• Step 3: Write the linear equation.

Let's work through each step:

Step 1: Calculate the slope $m$.
Using the points $(0, 5)$ and $(1, 6)$, the slope $m$ is calculated as follows:

$m = \frac{6 - 5}{1 - 0} = \frac{1}{1} = 1$

Step 2: Use the slope ($m = 1$) to find the $y$-intercept $b$.
We know from the point $(0, 5)$ that when $x = 0$, $y = 5$, which directly gives us the $y$-intercept:

$b = 5$

Step 3: Form the equation of the line using $y = mx + b$.
Substitute the found values into the equation:

$y = 1 \cdot x + 5$

Simplifying gives:

$y = x + 5$

Thus, the equation corresponding to the function is $y = x + 5$.

To find the equation of the linear function corresponding to the given table, follow these steps:

• Step 1: Identify Points
  We have three points from the table: $(-2, 4)$, $(-1, 2)$, and $(1, -2)$.
• Step 2: Calculate the Slope
  The slope $m$ can be calculated using any two points. Choosing $(-2, 4)$ and $(-1, 2)$:
  $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 4}{-1 - (-2)} = \frac{-2}{1} = -2$.
• Step 3: Verify Linear Relationship
  Using $(x, y)$ pairs, check another pair such as $(-1, 2)$ and $(1, -2)$:
  $m = \frac{-2 - 2}{1 - (-1)} = \frac{-4}{2} = -2$.
• Step 4: Select the Correct Equation
  The slope is $-2$, and since a linear function typically can be formatted as $y = mx + b$, we can see if $b = 0$ by trying one of the equations $y = -2x$ given in the choices. Let’s check:
  For $x = -2$: $y = -2(-2) = 4$. Matches.
  For $x = -1$: $y = -2(-1) = 2$. Matches.
  For $x = 1$: $y = -2(1) = -2$. Matches.

Therefore, the equation that corresponds to the function is $y = -2x$.

To determine the equation of the linear function from the given table, we'll follow these steps:

• Step 1: Calculate the Slope ($m$)
• Step 2: Find the Y-intercept ($b$)
• Step 3: Formulate the Linear Equation

Let's delve into each step:

Step 1: Calculate the Slope ($m$)
Using two points from the table, $(2, 13)$ and $(1, 10)$, we calculate the slope $m$ using the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 13}{1 - 2} = \frac{-3}{-1} = 3$

Thus, the slope $m$ is 3.

Step 2: Find the Y-intercept ($b$)
The y-intercept $b$ is easily found because it is where $x = 0$. From the table, when $x = 0$, $y = 7$, therefore $b = 7$.

Step 3: Formulate the Linear Equation
Substitute the values of $m$ and $b$ into the linear equation format:

$y = 3x + 7$

Hence, the equation that represents the linear function is $y = 3x + 7$.

Checking against the provided choices, the equation corresponds to choice 2.