Linear Function y=mx+b: Increase and decrease in relation to slope

Let's solve the problem by recognizing how the slope influences a linear function:

• Recall the form of a linear function: $y = mx + b$.
• The slope $m$ defines the inclination of the line:
  • Positive slope ($m > 0$): Function increases as $x$ increases.
  • Negative slope ($m < 0$): Function decreases as $x$ increases.
  • Zero slope ($m = 0$): Function is constant, forms a horizontal line.

Given the question is about a negative slope, we focus on the behavior when $m < 0$. In such a case, as $x$ increases, the value of $y$ decreases. This is because the line slopes downward from left to right.

Therefore, with a negative slope, the graph of the linear function is decreasing.

Hence, the correct answer is Choice 3: If the slope is negative then the function is decreasing.

To solve this problem, we'll examine the statements related to the slope of a linear function and determine which are true:

• A linear function is described mathematically by the equation $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• The slope $m$ determines the direction of the line:

  • If m > 0 , the line is increasing as $x$ increases.

  • If $m = 0$, the line is horizontal, meaning it is constant.

  • If m < 0 , the line is decreasing as $x$ increases.

Now, let's match these characteristics to the provided choices:

• If the slope is positive, then the function is increasing.

  • This is true as per the description above; a positive slope means the function increases as $x$ increases.

• If the slope is negative, then the function is constant.

  • This is incorrect; a negative slope results in a decreasing function.

• If the slope is positive, then the function is decreasing.

  • This is incorrect; a positive slope corresponds to an increasing function.

• If the slope is negative, then the function is increasing.

  • This is incorrect; a negative slope means the function is decreasing.

Therefore, the correct statement is that If the slope is positive, then the function is increasing.

To solve this problem, we need to analyze the behavior of a linear function when subject to various slope conditions.

We'll use the slope-intercept form of a line: $y = mx + b$, where $m$ denotes the slope:

• If $m = 0$: The equation becomes $y = b$. This equation describes a horizontal line that does not change as $x$ changes. Thus, the function is constant, matching choice 2.
• If $m > 0$: The line increases as $x$ increases, implying the function is increasing. This is not directly related to the condition of $m = 0$.
• If $m < 0$: The line decreases as $x$ increases, implying the function is decreasing. This is also not directly related to the condition of $m = 0$.

Choice 2 states that "If the slope is zero, then the function is constant." This is the correct statement because a zero slope indicates a horizontal line with no change in value of $y$ as the independent variable $x$ changes.

Therefore, the correct choice is: Choice 2: If the slope is zero, then the function is constant.

Let's identify the nature of the function given the points $B(0,0)$ and $A(2,0)$.

Step 1: Calculate the slope $m$ of the line using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

The coordinates given are $B(0,0)$ and $A(2,0)$. Here, $x_1 = 0$, $y_1 = 0$, $x_2 = 2$, and $y_2 = 0$.

Plug these values into the formula:

$m = \frac{0 - 0}{2 - 0} = \frac{0}{2} = 0$

The slope $m$ is $0$.

Step 2: Write the linear function equation.

Using the equation of a line in slope-intercept form $y = mx + b$, where $m = 0$:

$y = 0x + b$

Since both points $B(0,0)$ and $A(2,0)$ satisfy $y = 0$, the y-intercept $b = 0$.

Thus, the equation of the line is $y = 0$.

This equation represents a constant function, specifically the x-axis, where $y$ remains constant at zero for any $x$.

Therefore, the nature of the function given the points B(0,0) and A(2,0) is a constant function.

To determine the type of linear function represented by a line passing through the points $B(0,0)$ and $A(4,6)$, we follow these steps:

• Step 1: Identify the points. Here, we have $B(0,0)$ and $A(4,6)$.
• Step 2: Use the slope formula to find the slope of the line. The formula to find the slope $m$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Plug in the coordinates of the points:

$m = \frac{6 - 0}{4 - 0} = \frac{6}{4} = \frac{3}{2}$.

• Step 3: Analyze the slope: Since the slope $m = \frac{3}{2}$ is positive, the function is an increasing function.
• Step 4: Determine the type of function: A positive slope indicates that the function is increasing as we move from left to right on the graph.

Therefore, the correct description of this linear function, based on the given options, is a Bottom-up function.

To solve this problem, follow these steps:

• Step 1: Identify the points given: $B(0,5)$ and $A(6,5)$.
• Step 2: Use the slope formula to calculate the slope $m$ between these points:

The slope $m$ is calculated as:

Since the slope $m = 0$, the line is horizontal.

• Step 3: Classify the line as a "constant function" because the slope is zero, indicating no increase or decrease.

The line is therefore a constant function where the $y$-value remains at 5 for all $x$-values.

Therefore, the answer is constant function.

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

• Step 1: Calculate the slope using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
  Given points are $B(4,7)$ and $A(7,2)$. Therefore, $x_1 = 4$, $y_1 = 7$, $x_2 = 7$, $y_2 = 2$.
• Step 2: Substitute the values into the formula:
  $m = \frac{2 - 7}{7 - 4} = \frac{-5}{3} = -\frac{5}{3}$.
• Step 3: Determine the nature of the function based on the slope:
  Since $m = -\frac{5}{3}$ is negative, the function is a decreasing function.

Therefore, based on the slope being negative, the function represented by the line passing through these points is a Decreasing function.

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

• Step 1: Identify the given points $B(1, 5)$ and $A(7, 1)$.
• Step 2: Calculate the slope using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• Step 3: Interpret the result to determine if the function is increasing, decreasing, or constant.

Now, let's work through each step:
Step 1: The given points are $B(1, 5)$ and $A(7, 1)$.
Step 2: We calculate the slope as follows: $m = \frac{1 - 5}{7 - 1} = \frac{-4}{6} = -\frac{2}{3}$ Step 3: The slope $m = -\frac{2}{3}$ is negative, indicating that the function decreases as $x$ increases.

Therefore, since the slope is negative, the function is a decreasing function.

The correct choice is the first option: Decreasing function.

To determine the nature of the linear function, let's calculate the slope of the line passing through the given points:

• Given points: $(x_1, y_1) = \left(\frac{1}{2}, 5\right)$ and $(x_2, y_2) = \left(5\frac{1}{2}, 10\right)$.
• Use the formula for the slope: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 5}{5.5 - 0.5} = \frac{5}{5} = 1.$
• The slope $m = 1$ is positive, indicating the line is increasing from left to right.

Hence, the function is a bottom-up function, indicating it increases as the x-values increase.

Therefore, the correct answer is: Bottom-up function.

To solve this problem, we will compute the slope of the line that passes through the points $A(6, 30)$ and $B(36, -60)$.

Step 1: Apply the slope formula
The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is computed as follows:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the values for points $A(6, 30)$ and $B(36, -60)$:
$m = \frac{-60 - 30}{36 - 6} = \frac{-90}{30} = -3$

Step 2: Analyze the slope
Since the slope $m = -3$ is negative, it indicates that the linear function is decreasing.

Therefore, the solution to the problem is a decreasing function.