If $y=2x+1$ Were Changed To $y=\frac{1}{2}x+1$, How Would The Graph Of The New Function Compare With The First One?A. It Would Be Shifted Left. B. It Would Be Shifted Down. C. It Would Be Steeper. D. It Would Be Less Steep.
Introduction
In mathematics, linear functions are a fundamental concept in algebra and graphing. A linear function is a polynomial function of degree one, which can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. In this article, we will explore how changing the slope of a linear function affects its graph.
The Original Function
The original function is given by y = 2x + 1. This function has a slope of 2 and a y-intercept of 1. The graph of this function is a straight line that passes through the point (0, 1) and has a slope of 2.
The New Function
The new function is given by y = (1/2)x + 1. This function has a slope of 1/2 and a y-intercept of 1. The graph of this function is also a straight line that passes through the point (0, 1) but has a slope of 1/2.
Comparing the Graphs
To compare the graphs of the two functions, we need to consider the effect of changing the slope from 2 to 1/2. When the slope is increased, the graph becomes steeper, and when the slope is decreased, the graph becomes less steep.
In this case, the slope of the new function is 1/2, which is less than the slope of the original function. Therefore, the graph of the new function will be less steep than the graph of the original function.
Shifting the Graph
Shifting the graph of a function means moving it horizontally or vertically. In this case, the graph of the new function is not shifted horizontally or vertically. The y-intercept of both functions is the same, which means that the graph of the new function is not shifted down.
Conclusion
In conclusion, changing the slope of a linear function from 2 to 1/2 results in a graph that is less steep than the original graph. The graph of the new function is not shifted horizontally or vertically, and the y-intercept remains the same.
Answer
The correct answer is D. It would be less steep.
Key Takeaways
- Changing the slope of a linear function affects its graph.
- Increasing the slope makes the graph steeper, while decreasing the slope makes the graph less steep.
- The y-intercept of a linear function remains the same when the slope is changed.
Example Problems
- If y = 3x + 2 were changed to y = (1/3)x + 2, how would the graph of the new function compare with the first one?
- If y = 2x - 1 were changed to y = (1/2)x - 1, how would the graph of the new function compare with the first one?
Solutions
- The graph of the new function would be less steep than the graph of the original function.
- The graph of the new function would be less steep than the graph of the original function.
Practice Problems
- If y = 4x + 3 were changed to y = (1/4)x + 3, how would the graph of the new function compare with the first one?
- If y = 2x + 2 were changed to y = (1/2)x + 2, how would the graph of the new function compare with the first one?
Solutions
- The graph of the new function would be less steep than the graph of the original function.
- The graph of the new function would be less steep than the graph of the original function.
Q&A: Comparing the Graphs of Linear Functions =====================================================
Q: What happens to the graph of a linear function when the slope is increased?
A: When the slope of a linear function is increased, the graph becomes steeper. This means that for a given change in x, the change in y is greater.
Q: What happens to the graph of a linear function when the slope is decreased?
A: When the slope of a linear function is decreased, the graph becomes less steep. This means that for a given change in x, the change in y is smaller.
Q: How does the y-intercept affect the graph of a linear function?
A: The y-intercept of a linear function is the point where the graph intersects the y-axis. Changing the y-intercept shifts the graph vertically, but does not affect the slope.
Q: What is the effect of changing the slope and y-intercept on the graph of a linear function?
A: Changing the slope affects the steepness of the graph, while changing the y-intercept affects the vertical position of the graph. The two changes can be combined to create a new graph that is both steeper and shifted vertically.
Q: How can you determine the effect of changing the slope on the graph of a linear function?
A: To determine the effect of changing the slope, compare the slopes of the original and new functions. If the new slope is greater than the original slope, the graph will be steeper. If the new slope is less than the original slope, the graph will be less steep.
Q: What is the relationship between the slope and the graph of a linear function?
A: The slope of a linear function determines the steepness of the graph. A greater slope results in a steeper graph, while a smaller slope results in a less steep graph.
Q: Can the slope of a linear function be negative?
A: Yes, the slope of a linear function can be negative. A negative slope indicates that the graph slopes downward from left to right.
Q: What is the effect of a negative slope on the graph of a linear function?
A: A negative slope causes the graph to slope downward from left to right. This means that as x increases, y decreases.
Q: Can the slope of a linear function be zero?
A: Yes, the slope of a linear function can be zero. A slope of zero indicates that the graph is a horizontal line.
Q: What is the effect of a slope of zero on the graph of a linear function?
A: A slope of zero causes the graph to be a horizontal line. This means that y remains constant for all values of x.
Q: How can you determine the effect of changing the slope on the graph of a linear function?
A: To determine the effect of changing the slope, compare the slopes of the original and new functions. If the new slope is greater than the original slope, the graph will be steeper. If the new slope is less than the original slope, the graph will be less steep.
Q: What is the relationship between the slope and the graph of a linear function?
A: The slope of a linear function determines the steepness of the graph. A greater slope results in a steeper graph, while a smaller slope results in a less steep graph.
Q: Can the slope of a linear function be a fraction?
A: Yes, the slope of a linear function can be a fraction. A fraction as a slope indicates that the graph slopes upward or downward at a rate that is a fraction of the original slope.
Q: What is the effect of a fraction as a slope on the graph of a linear function?
A: A fraction as a slope causes the graph to slope upward or downward at a rate that is a fraction of the original slope. This means that for a given change in x, the change in y is smaller than the original change in y.