If Y = 2 X + 1 Y = 2x + 1 Y = 2 X + 1 Were Changed To Y = 1 2 X + 1 Y = \frac{1}{2}x + 1 Y = 2 1 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 Less Steep.D. It Would Be Steeper.
Introduction
In mathematics, linear functions are a fundamental concept in algebra and graphing. A linear function is a polynomial function of degree one, which means it has a single variable and a constant term. The general form of a linear function is y = mx + b, where m is the slope and b is the y-intercept. In this article, we will compare the graphs of two linear functions, y = 2x + 1 and y = (1/2)x + 1, and discuss how the change in the slope affects the graph.
Understanding the Original Function
The original function is y = 2x + 1. This function has a slope of 2, which means that for every unit increase in x, the value of y increases by 2 units. The y-intercept is 1, which means that the graph of the function passes through the point (0, 1). The graph of this function is a straight line with a positive slope.
Understanding the New Function
The new function is y = (1/2)x + 1. This function has a slope of 1/2, which means that for every unit increase in x, the value of y increases by 1/2 unit. The y-intercept is still 1, which means that the graph of the function still passes through the point (0, 1). However, the slope of the new function is half of the slope of the original function.
Comparing the Graphs
To compare the graphs of the two functions, we need to consider the effect of the change in the slope. When the slope of a linear function is reduced, the graph of the function becomes less steep. This is because the rate of change of the function is reduced, resulting in a more gradual increase in the value of y for a given increase in x.
In the case of the two functions, the graph of the new function, y = (1/2)x + 1, is less steep than the graph of the original function, y = 2x + 1. This is because the slope of the new function is half of the slope of the original function. As a result, the graph of the new function will have a more gradual increase in the value of y for a given increase in x.
Conclusion
In conclusion, if the function y = 2x + 1 were changed to y = (1/2)x + 1, the graph of the new function would be less steep than the graph of the original function. This is because the slope of the new function is half of the slope of the original function, resulting in a more gradual increase in the value of y for a given increase in x.
Key Takeaways
- The slope of a linear function determines the steepness of the graph.
- A reduction in the slope of a linear function results in a less steep graph.
- The y-intercept of a linear function determines the point at which the graph passes through the y-axis.
- The graph of a linear function with a positive slope is a straight line that increases as x increases.
Real-World Applications
The concept of linear functions and their graphs has many real-world applications. For example, in economics, the demand curve for a product is often modeled using a linear function. The slope of the demand curve represents the rate of change of the demand for the product in response to changes in the price. In physics, the motion of an object can be modeled using linear functions, where the slope of the function represents the velocity of the object.
Practice Problems
- If the function y = 3x + 2 were changed to y = (1/3)x + 2, how would the graph of the new function compare with the original function?
- If the function y = 2x - 1 were changed to y = (1/2)x - 1, how would the graph of the new function compare with the original function?
- If the function y = x + 1 were changed to y = (1/2)x + 1, how would the graph of the new function compare with the original function?
Answer Key
- 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.
- The graph of the new function would be less steep than the graph of the original function.
References
- [1] "Linear Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/linfunc.html
- [2] "Graphing Linear Functions" by Khan Academy. Retrieved from <https://www.khanacademy.org/math/algebra/x2f6f7f6/x2f6f7f7/x2f6f7f8/x2f6f7f9/x2f6f7fa/x2f6f7fb/x2f6f7fc/x2f6f7fd/x2f6f7fe/x2f6f7ff/x2f6f7fg/x2f6f7fh/x2f6f7fi/x2f6f7fj/x2f6f7fk/x2f6f7fl/x2f6f7fm/x2f6f7fn/x2f6f7fo/x2f6f7fp/x2f6f7fq/x2f6f7fr/x2f6f7fs/x2f6f7ft/x2f6f7fu/x2f6f7fv/x2f6f7fw/x2f6f7fx/x2f6f7fy/x2f6f7fz/x2f6f7ga/x2f6f7gb/x2f6f7gc/x2f6f7gd/x2f6f7ge/x2f6f7gf/x2f6f7gg/x2f6f7gh/x2f6f7gi/x2f6f7gj/x2f6f7gk/x2f6f7gl/x2f6f7gm/x2f6f7gn/x2f6f7go/x2f6f7gp/x2f6f7gq/x2f6f7gr/x2f6f7gs/x2f6f7gt/x2f6f7gu/x2f6f7gv/x2f6f7gw/x2f6f7gx/x2f6f7gy/x2f6f7gz/x2f6f7ha/x2f6f7hb/x2f6f7hc/x2f6f7hd/x2f6f7he/x2f6f7hf/x2f6f7hg/x2f6f7hh/x2f6f7hi/x2f6f7hj/x2f6f7hk/x2f6f7hl/x2f6f7hm/x2f6f7hn/x2f6f7ho/x2f6f7hp/x2f6f7hq/x2f6f7hr/x2f6f7hs/x2f6f7ht/x2f6f7hu/x2f6f7hv/x2f6f7hw/x2f6f7hx/x2f6f7hy/x2f6f7hz/x2f6f7ia/x2f6f7ib/x2f6f7ic/x2f6f7id/x2f6f7ie/x2f6f7if/x2f6f7ig/x2f6f7ih/x2f6f7ii/x2f6f7ij/x2f6f7ik/x2f6f7il/x2f6f7im/x2f6f7in/x2f6f7io/x2f6f7ip/x2f6f7iq/x2f6f7ir/x2f6f7is/x2f6f7it/x2f6f7iu/x2f6f7iv/x2f6f7iw/x2f6f7ix/x2f6f7iy/x2f6f7iz/x2f6f7ja/x2f6f7jb/x2f6f7jc/x2f6f7jd/x2f6f7je/x2f6f7jf/x2f6f7jg/x2f6f7jh/x2f6f7ji/x2f6f7jj/x2f6f7jk/x2f6f7jl/x2f6f7jm/x2f6f7jn/x2f6f7jo/x2f6f7jp/x2f6f7jq/x2f6f7jr/x2f6f7js/x2f6f7jt/x2f6f7ju/x2f6f7jv/x2f6f7jw/x2f6f7jx/x2f6f7jy/x2f6f7jz/x2f6f7ka/x2f6f7kb/x2f6f7kc/x2f6f7kd/x2f6f7ke/x2f6f7kf/x2f6f7kg/x2f6f7kh/x2f6f7ki/x2f6f7kj/x2f6f7kk/x2
Q&A: Comparing the Graphs of Two Linear Functions =====================================================
Q: What is the main difference between the two linear functions, y = 2x + 1 and y = (1/2)x + 1?
A: The main difference between the two linear functions is the slope. The original function, y = 2x + 1, has a slope of 2, while the new function, y = (1/2)x + 1, has a slope of 1/2.
Q: How does the change in the slope affect the graph of the new function?
A: The change in the slope affects the steepness of the graph. The new function, y = (1/2)x + 1, has a less steep graph than the original function, y = 2x + 1.
Q: What is the effect of reducing the slope of a linear function on the graph?
A: Reducing the slope of a linear function results in a less steep graph. This means that for a given increase in x, the value of y increases at a slower rate.
Q: How does the y-intercept affect the graph of a linear function?
A: The y-intercept determines the point at which the graph passes through the y-axis. In the case of the two functions, the y-intercept is 1, which means that both graphs pass through the point (0, 1).
Q: What is the relationship between the slope and the steepness of a linear function?
A: The slope of a linear function determines the steepness of the graph. A higher slope results in a steeper graph, while a lower slope results in a less steep graph.
Q: Can you give an example of a real-world application of linear functions?
A: Yes, one example of a real-world application of linear functions is in economics, where the demand curve for a product is often modeled using a linear function. The slope of the demand curve represents the rate of change of the demand for the product in response to changes in the price.
Q: How can you determine the slope of a linear function?
A: To determine the slope of a linear function, you can use the formula: slope = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the graph.
Q: What is the significance of the y-intercept in a linear function?
A: The y-intercept is significant because it determines the point at which the graph passes through the y-axis. This point is often used as a reference point to determine the slope and other characteristics of the graph.
Q: Can you explain the concept of linear functions in simple terms?
A: Yes, linear functions are a way of describing a relationship between two variables, x and y, using a straight line. The slope of the line determines the rate of change of y in response to changes in x.
Q: How can you use linear functions to model real-world situations?
A: You can use linear functions to model real-world situations by identifying the variables and the relationship between them. For example, you can use a linear function to model the cost of producing a product, where the cost is a function of the number of units produced.
Q: What are some common applications of linear functions in science and engineering?
A: Some common applications of linear functions in science and engineering include modeling population growth, predicting the motion of objects, and designing electrical circuits.
Q: Can you give an example of a linear function in a real-world context?
A: Yes, one example of a linear function in a real-world context is the cost of producing a product, where the cost is a function of the number of units produced. For example, if the cost of producing x units of a product is $2x + $100, then the cost is a linear function of the number of units produced.
Q: How can you use linear functions to solve problems in mathematics and science?
A: You can use linear functions to solve problems in mathematics and science by identifying the variables and the relationship between them. For example, you can use a linear function to model the motion of an object, where the position of the object is a function of time.
Q: What are some common mistakes to avoid when working with linear functions?
A: Some common mistakes to avoid when working with linear functions include:
- Failing to identify the slope and y-intercept of the function
- Using the wrong formula to calculate the slope
- Failing to check the units of the variables
- Failing to consider the domain and range of the function
Q: Can you explain the concept of linear functions in terms of a real-world example?
A: Yes, one example of a linear function in a real-world context is the cost of producing a product, where the cost is a function of the number of units produced. For example, if the cost of producing x units of a product is $2x + $100, then the cost is a linear function of the number of units produced.