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Sagot :
Let's analyze each statement to determine which features correctly describe the graphed function.
1. The equation of the function is [tex]\( y - 100 = -\frac{2}{25}(x - 8) \)[/tex].
To check this, we rearrange the given equation into slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 100 = -\frac{2}{25}(x - 8) \][/tex]
Distribute [tex]\(-\frac{2}{25}\)[/tex]:
[tex]\[ y - 100 = -\frac{2}{25}x + \frac{16}{25} \][/tex]
Simplify to:
[tex]\[ y = -\frac{2}{25}x + 100 + \frac{16}{25} \][/tex]
Since there's an issue with the simplification, the equation won't match our possible candidates. Therefore, statement 1 is incorrect.
2. The equation of the function is [tex]\( 25x + 2y = 200 \)[/tex].
We can solve for [tex]\( y \)[/tex] to identify the equation:
[tex]\[ 2y = -25x + 200 \][/tex]
Divide by 2:
[tex]\[ y = -\frac{25}{2}x + 100 \][/tex]
This matches one of our possible candidates. Hence, statement 2 is correct.
3. The [tex]\( y \)[/tex]-intercept for the function is [tex]\( (0,120) \)[/tex].
From the correct equation [tex]\( y = -\frac{25}{2}x + 100 \)[/tex], when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 100 \][/tex]
Therefore, the function intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0, 100) \)[/tex]. Thus, statement 3 is incorrect.
4. The [tex]\( x \)[/tex]-intercept for the function is [tex]\( (8, 0) \)[/tex].
To find the [tex]\( x \)[/tex]-intercept, set [tex]\( y = 0 \)[/tex] in the equation [tex]\( y = -\frac{25}{2}x + 100 \)[/tex]:
[tex]\[ 0 = -\frac{25}{2}x + 100 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ \frac{25}{2}x = 100 \][/tex]
Multiply both sides by [tex]\(\frac{2}{25} \)[/tex]:
[tex]\[ x = 8 \][/tex]
Therefore, the [tex]\( x \)[/tex]-intercept is indeed [tex]\( (8, 0) \)[/tex]. Statement 4 is correct.
5. The domain for the function is [tex]\([0, 12]\)[/tex].
The domain specifies the range of allowable [tex]\( x \)[/tex]-values. If the function represents the battery life during streaming, it makes logical sense to consider the domain within [tex]\( [0, 12] \)[/tex]. Hence, statement 5 is correct.
6. The equation of the function is [tex]\( y = -\frac{25}{2}x + 100 \)[/tex].
We verified this form from statement 2, so statement 6 is correct.
In summary, the correct statements are:
- The equation of the function is [tex]\( 25 x + 2 y = 200 \)[/tex].
- The [tex]\( x \)[/tex]-intercept for the function is [tex]\( (8, 0) \)[/tex].
- The domain for the function is [tex]\( [0,12] \)[/tex].
- The equation of the function is [tex]\( y = -\frac{25}{2}x + 100 \)[/tex].
1. The equation of the function is [tex]\( y - 100 = -\frac{2}{25}(x - 8) \)[/tex].
To check this, we rearrange the given equation into slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 100 = -\frac{2}{25}(x - 8) \][/tex]
Distribute [tex]\(-\frac{2}{25}\)[/tex]:
[tex]\[ y - 100 = -\frac{2}{25}x + \frac{16}{25} \][/tex]
Simplify to:
[tex]\[ y = -\frac{2}{25}x + 100 + \frac{16}{25} \][/tex]
Since there's an issue with the simplification, the equation won't match our possible candidates. Therefore, statement 1 is incorrect.
2. The equation of the function is [tex]\( 25x + 2y = 200 \)[/tex].
We can solve for [tex]\( y \)[/tex] to identify the equation:
[tex]\[ 2y = -25x + 200 \][/tex]
Divide by 2:
[tex]\[ y = -\frac{25}{2}x + 100 \][/tex]
This matches one of our possible candidates. Hence, statement 2 is correct.
3. The [tex]\( y \)[/tex]-intercept for the function is [tex]\( (0,120) \)[/tex].
From the correct equation [tex]\( y = -\frac{25}{2}x + 100 \)[/tex], when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 100 \][/tex]
Therefore, the function intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0, 100) \)[/tex]. Thus, statement 3 is incorrect.
4. The [tex]\( x \)[/tex]-intercept for the function is [tex]\( (8, 0) \)[/tex].
To find the [tex]\( x \)[/tex]-intercept, set [tex]\( y = 0 \)[/tex] in the equation [tex]\( y = -\frac{25}{2}x + 100 \)[/tex]:
[tex]\[ 0 = -\frac{25}{2}x + 100 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ \frac{25}{2}x = 100 \][/tex]
Multiply both sides by [tex]\(\frac{2}{25} \)[/tex]:
[tex]\[ x = 8 \][/tex]
Therefore, the [tex]\( x \)[/tex]-intercept is indeed [tex]\( (8, 0) \)[/tex]. Statement 4 is correct.
5. The domain for the function is [tex]\([0, 12]\)[/tex].
The domain specifies the range of allowable [tex]\( x \)[/tex]-values. If the function represents the battery life during streaming, it makes logical sense to consider the domain within [tex]\( [0, 12] \)[/tex]. Hence, statement 5 is correct.
6. The equation of the function is [tex]\( y = -\frac{25}{2}x + 100 \)[/tex].
We verified this form from statement 2, so statement 6 is correct.
In summary, the correct statements are:
- The equation of the function is [tex]\( 25 x + 2 y = 200 \)[/tex].
- The [tex]\( x \)[/tex]-intercept for the function is [tex]\( (8, 0) \)[/tex].
- The domain for the function is [tex]\( [0,12] \)[/tex].
- The equation of the function is [tex]\( y = -\frac{25}{2}x + 100 \)[/tex].
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