Slope: Logic and comprehension questions

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.

The function $y = -5x + 31$ is analyzed as follows:

• The slope of the function $y = -5x + 31$ is $-5$, which is negative. This implies that the function is decreasing. Therefore, option 3 is correct.
• The function $y = -5x + 31$ is in the form $y = mx + b$, with a slope ($m$) of $-5$. The function $y = 3 - 5x$ can be rewritten as $y = -5x + 3$, where the slope is also $-5$. Hence, these two lines are parallel. Therefore, option 1 is correct.
• The function $y = 2x + 31$ intersects the y-axis at $y = 31$. The given function $y = -5x + 31$ also intersects the y-axis at $y = 31$. Therefore, option 2 is correct.

Given that all individually assessed options are correct, the best answer is option 4: "All answers are correct."