Question
Determine if the relation y = 4x - 1 is a function by first graphing the relation, the applying the vertical line test.
Answers
GPT-4o mini
To determine if the relation \( y = 4x - 1 \) is a function, we can analyze it through graphing and applying the vertical line test.
1. **Graphing the relation**:
The equation \( y = 4x - 1 \) is a linear equation of the form \( y = mx + b \), where \( m \) (the slope) is 4 and \( b \) (the y-intercept) is -1. To graph the equation:
- Start by plotting the y-intercept (0, -1).
- Use the slope to find another point. Since the slope is 4, you can rise 4 units up for every 1 unit you move to the right. From (0, -1), move right 1 to (1, -1 + 4) = (1, 3).
- Connect these points with a straight line.
2. **Applying the vertical line test**:
The vertical line test states that if a vertical line intersects the graph of a relation at more than one point, then the relation is not a function.
- If you draw a vertical line (for example, \( x = 0 \) or any other vertical line) on the graph of \( y = 4x - 1 \), you will see that it intersects the line at only one point (in this case, at (0, -1)).
- Since every vertical line intersects the graph at one point, the relation passes the vertical line test.
Since the relation \( y = 4x - 1 \) meets the criteria of being a function (it passes the vertical line test), we can conclude that **yes, \( y = 4x - 1 \) is a function**.
1. **Graphing the relation**:
The equation \( y = 4x - 1 \) is a linear equation of the form \( y = mx + b \), where \( m \) (the slope) is 4 and \( b \) (the y-intercept) is -1. To graph the equation:
- Start by plotting the y-intercept (0, -1).
- Use the slope to find another point. Since the slope is 4, you can rise 4 units up for every 1 unit you move to the right. From (0, -1), move right 1 to (1, -1 + 4) = (1, 3).
- Connect these points with a straight line.
2. **Applying the vertical line test**:
The vertical line test states that if a vertical line intersects the graph of a relation at more than one point, then the relation is not a function.
- If you draw a vertical line (for example, \( x = 0 \) or any other vertical line) on the graph of \( y = 4x - 1 \), you will see that it intersects the line at only one point (in this case, at (0, -1)).
- Since every vertical line intersects the graph at one point, the relation passes the vertical line test.
Since the relation \( y = 4x - 1 \) meets the criteria of being a function (it passes the vertical line test), we can conclude that **yes, \( y = 4x - 1 \) is a function**.
you sure
GPT-4o mini
Yes, I'm sure! Let's clarify the reasoning:
1. **Definition of a Function**: A relation is a function if every input (or x-value) has exactly one output (or y-value).
2. **Equation**: The relation given is \( y = 4x - 1 \). This is a linear equation and represents a straight line. For every value of \( x \), you can calculate exactly one corresponding value of \( y\).
3. **Graphing**: When you plot the graph of this equation, as described earlier, you will see a straight line.
4. **Vertical Line Test**: If you draw any vertical line (which represents a specific x-value), it will intersect the graph of the equation at exactly one point. If that is true for all vertical lines drawn, then it confirms that the relation is indeed a function.
Since all these checks confirm that each x-value has exactly one y-value and that the vertical line test is passed, we can confidently conclude that \( y = 4x - 1 \) is a function.
If you have further questions or would like to discuss this in more detail, feel free to ask!
1. **Definition of a Function**: A relation is a function if every input (or x-value) has exactly one output (or y-value).
2. **Equation**: The relation given is \( y = 4x - 1 \). This is a linear equation and represents a straight line. For every value of \( x \), you can calculate exactly one corresponding value of \( y\).
3. **Graphing**: When you plot the graph of this equation, as described earlier, you will see a straight line.
4. **Vertical Line Test**: If you draw any vertical line (which represents a specific x-value), it will intersect the graph of the equation at exactly one point. If that is true for all vertical lines drawn, then it confirms that the relation is indeed a function.
Since all these checks confirm that each x-value has exactly one y-value and that the vertical line test is passed, we can confidently conclude that \( y = 4x - 1 \) is a function.
If you have further questions or would like to discuss this in more detail, feel free to ask!