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Sagot :
To determine which statements are true, let's analyze the linear functions and their characteristics in detail.
### Determining the Slope of Function A
For Function A, we have the given points: (-9, 1) and (-6, 2).
The formula to calculate the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points into the formula:
[tex]\[ m = \frac{2 - 1}{-6 - (-9)} = \frac{1}{3} \][/tex]
Thus, the slope of Function A is:
[tex]\[ \frac{1}{3} \approx 0.333 \][/tex]
### Determining the Slope and Y-Intercept of Function B
Function B is given by the equation [tex]\( y = 3x - 3 \)[/tex]. Here, the slope is the coefficient of [tex]\( x \)[/tex], and the y-intercept is the constant term.
Thus,
- The slope of Function B is:
[tex]\[ 3 \][/tex]
- The y-intercept of Function B is:
[tex]\[ -3 \][/tex]
### Determining the Y-Intercept of Function A
To find the y-intercept of Function A, we use the slope we already found and one of the points. Let's use the point (-9, 1).
The linear equation is in the form:
[tex]\[ y = mx + b \][/tex]
Substitute the point (-9, 1) and the slope [tex]\( \frac{1}{3} \)[/tex]:
[tex]\[ 1 = \left(\frac{1}{3}\right)(-9) + b \][/tex]
[tex]\[ 1 = -3 + b \][/tex]
[tex]\[ b = 4 \][/tex]
Thus, the y-intercept of Function A is:
[tex]\[ 4 \][/tex]
### Comparing the Slopes
- The slope of Function A ([tex]\( \approx 0.333 \)[/tex]) is less than the slope of Function B ([tex]\( 3 \)[/tex]).
### Comparing the Y-Intercepts
- The y-intercept of Function A ([tex]\( 4 \)[/tex]) is greater than the y-intercept of Function B ([tex]\( -3 \)[/tex]).
### Summarizing the True Statements
Based on the calculations:
1. The statement "The slope of Function A is greater than the slope of Function B" is false.
2. The statement "The slope of Function A is less than the slope of Function B" is true.
3. The statement "The y-intercept of Function A is greater than the y-intercept of Function B" is true.
4. The statement "The y-intercept of Function A is less than the y-intercept of Function B" is false.
Therefore, the true statements are:
- The slope of Function A is less than the slope of Function B.
- The y-intercept of Function A is greater than the y-intercept of Function B.
### Determining the Slope of Function A
For Function A, we have the given points: (-9, 1) and (-6, 2).
The formula to calculate the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points into the formula:
[tex]\[ m = \frac{2 - 1}{-6 - (-9)} = \frac{1}{3} \][/tex]
Thus, the slope of Function A is:
[tex]\[ \frac{1}{3} \approx 0.333 \][/tex]
### Determining the Slope and Y-Intercept of Function B
Function B is given by the equation [tex]\( y = 3x - 3 \)[/tex]. Here, the slope is the coefficient of [tex]\( x \)[/tex], and the y-intercept is the constant term.
Thus,
- The slope of Function B is:
[tex]\[ 3 \][/tex]
- The y-intercept of Function B is:
[tex]\[ -3 \][/tex]
### Determining the Y-Intercept of Function A
To find the y-intercept of Function A, we use the slope we already found and one of the points. Let's use the point (-9, 1).
The linear equation is in the form:
[tex]\[ y = mx + b \][/tex]
Substitute the point (-9, 1) and the slope [tex]\( \frac{1}{3} \)[/tex]:
[tex]\[ 1 = \left(\frac{1}{3}\right)(-9) + b \][/tex]
[tex]\[ 1 = -3 + b \][/tex]
[tex]\[ b = 4 \][/tex]
Thus, the y-intercept of Function A is:
[tex]\[ 4 \][/tex]
### Comparing the Slopes
- The slope of Function A ([tex]\( \approx 0.333 \)[/tex]) is less than the slope of Function B ([tex]\( 3 \)[/tex]).
### Comparing the Y-Intercepts
- The y-intercept of Function A ([tex]\( 4 \)[/tex]) is greater than the y-intercept of Function B ([tex]\( -3 \)[/tex]).
### Summarizing the True Statements
Based on the calculations:
1. The statement "The slope of Function A is greater than the slope of Function B" is false.
2. The statement "The slope of Function A is less than the slope of Function B" is true.
3. The statement "The y-intercept of Function A is greater than the y-intercept of Function B" is true.
4. The statement "The y-intercept of Function A is less than the y-intercept of Function B" is false.
Therefore, the true statements are:
- The slope of Function A is less than the slope of Function B.
- The y-intercept of Function A is greater than the y-intercept of Function B.
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