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:
- Identify the y-intercept: The y-intercept is \(-2\) (the point (0, -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.
- 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).
- 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.