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Sagot :
Let's analyze both functions to compare their properties.
1. Function 1: [tex]\( y = 10x - 30 \)[/tex]
- Slope: The slope ([tex]\( m \)[/tex]) is the coefficient of [tex]\( x \)[/tex], which is 10. Since the slope is positive (10), it indicates that this function is increasing.
- Y-intercept: The [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]) is the constant term when [tex]\( x = 0 \)[/tex]. For this function, it is -30. Therefore, the [tex]\( y \)[/tex]-intercept is at the point [tex]\( (0, -30) \)[/tex].
2. Function 2: This function is described as decreasing steadily at a rate of 2 and has an initial value of 6.
- Slope: The rate at which the function is decreasing is represented by a negative slope. In this case, the slope is -2, indicating that this function is decreasing.
- Y-intercept: The initial value, when [tex]\( x = 0 \)[/tex], is given as 6. Therefore, the [tex]\( y \)[/tex]-intercept is at the point [tex]\( (0, 6) \)[/tex].
Now we can correctly compare the two functions based on their properties:
Function 1:
- Slope: 10 (positive)
- [tex]\( Y \)[/tex]-intercept: -30
Function 2:
- Slope: -2 (negative)
- [tex]\( Y \)[/tex]-intercept: 6
Given these comparisons, let's evaluate the statements:
1. Function 1 and Function 2 both have a positive slope.
- This statement is false. Function 1 has a positive slope, but Function 2 has a negative slope.
2. Function 1 and Function 2 both have a negative slope.
- This statement is false. Function 1 has a positive slope and Function 2 has a negative slope.
3. Function 1 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, 6) \)[/tex], and Function 2 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, -30) \)[/tex].
- This statement is false. The intercept values are assigned to the wrong functions.
4. Function 1 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, -30) \)[/tex], and Function 2 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, 6) \)[/tex].
- This statement is true. Function 1 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, -30) \)[/tex], and Function 2 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, 6) \)[/tex].
Thus, the correct statement is:
Function 1 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, -30) \)[/tex], and Function 2 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, 6) \)[/tex].
1. Function 1: [tex]\( y = 10x - 30 \)[/tex]
- Slope: The slope ([tex]\( m \)[/tex]) is the coefficient of [tex]\( x \)[/tex], which is 10. Since the slope is positive (10), it indicates that this function is increasing.
- Y-intercept: The [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]) is the constant term when [tex]\( x = 0 \)[/tex]. For this function, it is -30. Therefore, the [tex]\( y \)[/tex]-intercept is at the point [tex]\( (0, -30) \)[/tex].
2. Function 2: This function is described as decreasing steadily at a rate of 2 and has an initial value of 6.
- Slope: The rate at which the function is decreasing is represented by a negative slope. In this case, the slope is -2, indicating that this function is decreasing.
- Y-intercept: The initial value, when [tex]\( x = 0 \)[/tex], is given as 6. Therefore, the [tex]\( y \)[/tex]-intercept is at the point [tex]\( (0, 6) \)[/tex].
Now we can correctly compare the two functions based on their properties:
Function 1:
- Slope: 10 (positive)
- [tex]\( Y \)[/tex]-intercept: -30
Function 2:
- Slope: -2 (negative)
- [tex]\( Y \)[/tex]-intercept: 6
Given these comparisons, let's evaluate the statements:
1. Function 1 and Function 2 both have a positive slope.
- This statement is false. Function 1 has a positive slope, but Function 2 has a negative slope.
2. Function 1 and Function 2 both have a negative slope.
- This statement is false. Function 1 has a positive slope and Function 2 has a negative slope.
3. Function 1 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, 6) \)[/tex], and Function 2 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, -30) \)[/tex].
- This statement is false. The intercept values are assigned to the wrong functions.
4. Function 1 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, -30) \)[/tex], and Function 2 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, 6) \)[/tex].
- This statement is true. Function 1 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, -30) \)[/tex], and Function 2 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, 6) \)[/tex].
Thus, the correct statement is:
Function 1 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, -30) \)[/tex], and Function 2 has a [tex]\( y \)[/tex]-intercept at point [tex]\( (0, 6) \)[/tex].
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