Question
all real numbers example, for a parabola
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GPT-4o mini
A parabola is a curve defined by a quadratic function, typically in the standard form:
\[ y = ax^2 + bx + c \]
where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).
To illustrate a parabola that can take all real numbers as outputs (the range), we can construct an example where the parabola opens upwards (i.e., \( a > 0 \)). Let’s take \( a = 1 \), \( b = 0 \), and \( c = 0 \). The function simplifies to:
\[ y = x^2 \]
### Properties of the Parabola \( y = x^2 \):
1. **Vertex**: The vertex of this parabola is at the point \( (0, 0) \).
2. **Direction**: Since the coefficient \( a = 1 \) is positive, the parabola opens upwards.
3. **Range**: The output (i.e., \( y \)) can take on any value from \( 0 \) to \( \infty \). Thus the range is \( [0, \infty) \).
4. **Domain**: The input (i.e., \( x \)) can be any real number. Thus the domain is \( (-\infty, \infty) \).
### Example of a Parabola that Covers All Real Numbers:
To create a parabola that outputs all real numbers, we can modify our approach slightly. Consider the function:
\[ y = x^2 - 1 \]
### Properties of the Parabola \( y = x^2 - 1 \):
1. **Vertex**: The vertex of this parabola is at the point \( (0, -1) \).
2. **Direction**: It still opens upwards.
3. **Range**: The output starts from \( -1 \) (the lowest point of the parabola) and goes to \( \infty \). Thus the range is \( [-1, \infty) \).
However, if we want a parabola that takes on all real numbers as outputs, we can use the following transformation:
### Example for a Parabola Covering All Real Numbers:
To create a function that gives all real outputs, one approach is:
\[ y = - (x^2) \]
### Properties of the Parabola \( y = - x^2 \):
1. **Vertex**: The vertex of this parabola is at the point \( (0, 0) \).
2. **Direction**: Since \( a = -1 \), the parabola opens downwards.
3. **Range**: The output (i.e., \( y \)) can take on any value from \( -\infty \) to \( 0 \). Thus the range is \( (-\infty, 0] \).
4. **Domain**: The input (i.e., \( x \)) can still be any real number. Thus the domain is \( (-\infty, \infty) \).
In conclusion: A simple example that gives all real values as outputs is \( y = -x^2 \), covering the whole range from \( -\infty \) to \( 0 \), effectively demonstrating a parabola with a downward opening. For a parabola to yield all real outputs positively we can't achieve with conventional parabolic shapes since quadratics are bounded based on their vertex orientation.
\[ y = ax^2 + bx + c \]
where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).
To illustrate a parabola that can take all real numbers as outputs (the range), we can construct an example where the parabola opens upwards (i.e., \( a > 0 \)). Let’s take \( a = 1 \), \( b = 0 \), and \( c = 0 \). The function simplifies to:
\[ y = x^2 \]
### Properties of the Parabola \( y = x^2 \):
1. **Vertex**: The vertex of this parabola is at the point \( (0, 0) \).
2. **Direction**: Since the coefficient \( a = 1 \) is positive, the parabola opens upwards.
3. **Range**: The output (i.e., \( y \)) can take on any value from \( 0 \) to \( \infty \). Thus the range is \( [0, \infty) \).
4. **Domain**: The input (i.e., \( x \)) can be any real number. Thus the domain is \( (-\infty, \infty) \).
### Example of a Parabola that Covers All Real Numbers:
To create a parabola that outputs all real numbers, we can modify our approach slightly. Consider the function:
\[ y = x^2 - 1 \]
### Properties of the Parabola \( y = x^2 - 1 \):
1. **Vertex**: The vertex of this parabola is at the point \( (0, -1) \).
2. **Direction**: It still opens upwards.
3. **Range**: The output starts from \( -1 \) (the lowest point of the parabola) and goes to \( \infty \). Thus the range is \( [-1, \infty) \).
However, if we want a parabola that takes on all real numbers as outputs, we can use the following transformation:
### Example for a Parabola Covering All Real Numbers:
To create a function that gives all real outputs, one approach is:
\[ y = - (x^2) \]
### Properties of the Parabola \( y = - x^2 \):
1. **Vertex**: The vertex of this parabola is at the point \( (0, 0) \).
2. **Direction**: Since \( a = -1 \), the parabola opens downwards.
3. **Range**: The output (i.e., \( y \)) can take on any value from \( -\infty \) to \( 0 \). Thus the range is \( (-\infty, 0] \).
4. **Domain**: The input (i.e., \( x \)) can still be any real number. Thus the domain is \( (-\infty, \infty) \).
In conclusion: A simple example that gives all real values as outputs is \( y = -x^2 \), covering the whole range from \( -\infty \) to \( 0 \), effectively demonstrating a parabola with a downward opening. For a parabola to yield all real outputs positively we can't achieve with conventional parabolic shapes since quadratics are bounded based on their vertex orientation.