Response area Linear Function Graphs Explained (Sample Questions)

Answer 1

Certainly! Below are some sample questions regarding linear function graphs, along with explanations and answers.

Sample Question 1: Identifying a Linear Function

Question: Which of the following equations represents a linear function?
A) $y=2x+3$
B) $y=x^2+2$
C) $y=3\sqrt{x}+1$
D) $y=-4x+5$

Answer:
A) and D) are linear functions. A linear function has the form $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.

• Option A: $y=2x+3$ is linear.
• Option B: $y=x^2+2$ is quadratic and not linear.
• Option C: $y=3\sqrt{x}+1$ is not linear because of the square root function.
• Option D: $y=-4x+5$ is also linear.

Sample Question 2: Finding the Slope and Y-Intercept

Question: What is the slope and y-intercept of the function given by the equation $y=-3x+7$?

Answer:
In the equation $y=mx+b$, $m$ represents the slope and $b$ represents the y-intercept.

• Here, $m=-3$ (slope)
• $b=7$ (y-intercept)

Sample Question 3: Graphing a Linear Equation

Question: Graph the linear equation $y=\frac{1}{2}x-2$.

Steps to Graph:

1. Identify the y-intercept: The y-intercept is $-2$ (the point (0, -2)).
2. Identify the slope: The slope is $\frac{1}{2}$, which means for every 1 unit you move up in the y-direction, you move 2 units in the x-direction.
3. From the y-intercept (0, -2), you can plot another point. Moving up 1 unit and 2 units to the right, you reach the point (2, -1).
4. Draw a straight line through the two points.

Sample Question 4: Graph Interpretation

Question: If the line in the graph of the equation $y=4x-1$ is plotted on a coordinate plane, what would be the expected behavior of the function as $x$ increases?

Answer:
As $x$ increases, the value of $y$ will also increase because the slope of the line is positive (4). This indicates that the line rises from left to right. Specifically, for every 1 unit increase in $x$, $y$ increases by 4 units.

Sample Question 5: Parallel Lines

Question: Which of the following lines is parallel to the line represented by $y=-2x+4$?
A) $y=-2x+1$
B) $y=3x-5$
C) $y=-\frac{1}{2}x+2$
D) $y=2x+4$

Answer:
Lines are parallel if they have the same slope. The slope of the line $y=-2x+4$ is $-2$.

• A) $y=-2x+1$ has the same slope $-2$ and is parallel.
• B) $y=3x-5$ has a slope of $3$, not parallel.
• C) $y=-\frac{1}{2}x+2$ has a slope of $-\frac{1}{2}$, not parallel.
• D) $y=2x+4$ has a slope of $2$, not parallel.
  Thus, only A) is parallel.

These sample questions cover a variety of concepts related to linear functions and graphs, providing a well-rounded understanding of the topic.