Question
Question 5
1 pts
Suppose f(x) = x and g(x) = 4x2
-. Which
statement best compares the graph of g(x) with the graph of f(x)?
O The graph of g(x) is the graph of f(x) shifted
4 units left.
O The graph of g(x) is the graph of f(x) vertically compressed by a factor of 4.
O The graph of g(x) is the graph of f(x) horizontally stretched by a factor of 4.
O The graph of g(x) is the graph of f(x) vertically stretched by a factor of 4.
1 pts
Suppose f(x) = x and g(x) = 4x2
-. Which
statement best compares the graph of g(x) with the graph of f(x)?
O The graph of g(x) is the graph of f(x) shifted
4 units left.
O The graph of g(x) is the graph of f(x) vertically compressed by a factor of 4.
O The graph of g(x) is the graph of f(x) horizontally stretched by a factor of 4.
O The graph of g(x) is the graph of f(x) vertically stretched by a factor of 4.
Answers
GPT-4o mini
To compare the graphs of \( f(x) = x \) and \( g(x) = 4x^2 \), let's analyze the transformations involved.
- The function \( f(x) = x \) is a linear function, which represents a straight line that passes through the origin with a slope of 1.
- The function \( g(x) = 4x^2 \) is a quadratic function, which represents a parabola that opens upwards.
Now, looking specifically at the transformation:
1. The basic form of a parabola is \( y = x^2 \).
2. In the function \( g(x) = 4x^2 \), the coefficient of 4 indicates that the parabola will be stretched vertically by a factor of 4 compared to the basic parabola \( y = x^2 \).
Thus, comparing the function \( g(x) = 4x^2 \) to the linear function \( f(x) = x \), it is clear that \( g(x) \) is a completely different function type. However, if we were to specifically compare vertical stretching of the basic parabola \( y = x^2 \) (which is not our main function \( f(x) = x \)), we would say that \( g(x) \) is vertically stretched.
However, in the context of the options given:
- **The correct interpretation should focus on vertical behavior**. The statement that best describes \( g(x) = 4x^2 \) in relation to the original linear function \( f(x) = x \) is that **it is not a direct transformation of \( f(x) = x\)**—so technically, none of the options perfectly apply since they refer to transformations of \( f(x) \) rather than comparing graph shapes.
But if we seek out the conceptual understanding, we can consider the stretching:
- From qualitative understanding and examining values from typical points:
- At \( x=1 \), \( f(1)=1 \) and \( g(1)=4 \)
- At \( x=2 \), \( f(2)=2 \) and \( g(2)=16 \)
Thus, \( g(x) \) indeed **shows more rapid growth than f(x)**.
The closest correct interpretation concerning the intended question could also imply possible misunderstanding in transformations, but the most accurate answer from those given, concerning what’s clearly true about vertical stretch properties of the curve alone applicable here is:
**The graph of \( g(x) \) is the graph of \( f(x) \) vertically stretched by a factor of 4** (referring to the implicit transformation nature).
- The function \( f(x) = x \) is a linear function, which represents a straight line that passes through the origin with a slope of 1.
- The function \( g(x) = 4x^2 \) is a quadratic function, which represents a parabola that opens upwards.
Now, looking specifically at the transformation:
1. The basic form of a parabola is \( y = x^2 \).
2. In the function \( g(x) = 4x^2 \), the coefficient of 4 indicates that the parabola will be stretched vertically by a factor of 4 compared to the basic parabola \( y = x^2 \).
Thus, comparing the function \( g(x) = 4x^2 \) to the linear function \( f(x) = x \), it is clear that \( g(x) \) is a completely different function type. However, if we were to specifically compare vertical stretching of the basic parabola \( y = x^2 \) (which is not our main function \( f(x) = x \)), we would say that \( g(x) \) is vertically stretched.
However, in the context of the options given:
- **The correct interpretation should focus on vertical behavior**. The statement that best describes \( g(x) = 4x^2 \) in relation to the original linear function \( f(x) = x \) is that **it is not a direct transformation of \( f(x) = x\)**—so technically, none of the options perfectly apply since they refer to transformations of \( f(x) \) rather than comparing graph shapes.
But if we seek out the conceptual understanding, we can consider the stretching:
- From qualitative understanding and examining values from typical points:
- At \( x=1 \), \( f(1)=1 \) and \( g(1)=4 \)
- At \( x=2 \), \( f(2)=2 \) and \( g(2)=16 \)
Thus, \( g(x) \) indeed **shows more rapid growth than f(x)**.
The closest correct interpretation concerning the intended question could also imply possible misunderstanding in transformations, but the most accurate answer from those given, concerning what’s clearly true about vertical stretch properties of the curve alone applicable here is:
**The graph of \( g(x) \) is the graph of \( f(x) \) vertically stretched by a factor of 4** (referring to the implicit transformation nature).