Answer :
To determine which function is increasing at the highest rate, we need to find the rate of change (slope) of each given option. Here's how we do it step-by-step:
### Option B:
Given the equation [tex]\( 12x - 6y = -24 \)[/tex].
First, we need to rewrite this equation in slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
1. Start with the given equation:
[tex]\[ 12x - 6y = -24 \][/tex]
2. Isolate [tex]\( y \)[/tex] on one side of the equation:
[tex]\[ -6y = -12x - 24 \][/tex]
3. Divide every term by -6:
[tex]\[ y = 2x + 4 \][/tex]
So, the slope ([tex]\( m \)[/tex]) of the function in Option B is:
[tex]\[ m = 2 \][/tex]
### Option C:
Given a table of values for [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline g(x) & -5 & -4 & -3 & -2 & -1 \\ \hline \end{array} \][/tex]
To find the rate of change, we calculate the slope using the difference in [tex]\( g(x) \)[/tex] values and the corresponding [tex]\( x \)[/tex] values.
1. Calculate the change in [tex]\( g(x) \)[/tex], which is the difference between the last and first values:
[tex]\[ g(x_5) - g(x_1) = -1 - (-5) = -1 + 5 = 4 \][/tex]
2. Calculate the change in [tex]\( x \)[/tex], which is the difference between the last and first [tex]\( x \)[/tex] values:
[tex]\[ x_5 - x_1 = 5 - 1 = 4 \][/tex]
3. The rate of change (slope) is:
[tex]\[ \text{slope} = \frac{4}{4} = 1 \][/tex]
So, the rate of change for [tex]\( g(x) \)[/tex] in Option C is:
[tex]\[ \text{slope} = 1 \][/tex]
### Option D:
A linear function [tex]\( f \)[/tex] with [tex]\( x \)[/tex]-intercept of 8 and [tex]\( y \)[/tex]-intercept of -4.
The [tex]\( x \)[/tex]-intercept is the point where the function crosses the x-axis ([tex]\( x = 8, y = 0 \)[/tex]) and the [tex]\( y \)[/tex]-intercept is the point where it crosses the y-axis ([tex]\( x = 0, y = -4 \)[/tex]).
To find the rate of change, we calculate the slope using these points:
1. Let [tex]\( (x_1, y_1) = (8, 0) \)[/tex] and [tex]\( (x_2, y_2) = (0, -4) \)[/tex]
2. The slope ([tex]\( m \)[/tex]) is calculated by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 0}{0 - 8} = \frac{-4}{-8} = \frac{1}{2} = 0.5 \][/tex]
So, the rate of change for [tex]\( f \)[/tex] in Option D is:
[tex]\[ m = 0.5 \][/tex]
### Comparison:
We have determined the slopes of the functions as follows:
- Option B: [tex]\( m = 2 \)[/tex]
- Option C: [tex]\( m = 1 \)[/tex]
- Option D: [tex]\( m = 0.5 \)[/tex]
Among these, the highest rate of change is [tex]\( 2 \)[/tex], which corresponds to Option B.
### Conclusion:
The function [tex]\( 12x - 6y = -24 \)[/tex] in Option B is increasing at the highest rate.
### Option B:
Given the equation [tex]\( 12x - 6y = -24 \)[/tex].
First, we need to rewrite this equation in slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
1. Start with the given equation:
[tex]\[ 12x - 6y = -24 \][/tex]
2. Isolate [tex]\( y \)[/tex] on one side of the equation:
[tex]\[ -6y = -12x - 24 \][/tex]
3. Divide every term by -6:
[tex]\[ y = 2x + 4 \][/tex]
So, the slope ([tex]\( m \)[/tex]) of the function in Option B is:
[tex]\[ m = 2 \][/tex]
### Option C:
Given a table of values for [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline g(x) & -5 & -4 & -3 & -2 & -1 \\ \hline \end{array} \][/tex]
To find the rate of change, we calculate the slope using the difference in [tex]\( g(x) \)[/tex] values and the corresponding [tex]\( x \)[/tex] values.
1. Calculate the change in [tex]\( g(x) \)[/tex], which is the difference between the last and first values:
[tex]\[ g(x_5) - g(x_1) = -1 - (-5) = -1 + 5 = 4 \][/tex]
2. Calculate the change in [tex]\( x \)[/tex], which is the difference between the last and first [tex]\( x \)[/tex] values:
[tex]\[ x_5 - x_1 = 5 - 1 = 4 \][/tex]
3. The rate of change (slope) is:
[tex]\[ \text{slope} = \frac{4}{4} = 1 \][/tex]
So, the rate of change for [tex]\( g(x) \)[/tex] in Option C is:
[tex]\[ \text{slope} = 1 \][/tex]
### Option D:
A linear function [tex]\( f \)[/tex] with [tex]\( x \)[/tex]-intercept of 8 and [tex]\( y \)[/tex]-intercept of -4.
The [tex]\( x \)[/tex]-intercept is the point where the function crosses the x-axis ([tex]\( x = 8, y = 0 \)[/tex]) and the [tex]\( y \)[/tex]-intercept is the point where it crosses the y-axis ([tex]\( x = 0, y = -4 \)[/tex]).
To find the rate of change, we calculate the slope using these points:
1. Let [tex]\( (x_1, y_1) = (8, 0) \)[/tex] and [tex]\( (x_2, y_2) = (0, -4) \)[/tex]
2. The slope ([tex]\( m \)[/tex]) is calculated by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 0}{0 - 8} = \frac{-4}{-8} = \frac{1}{2} = 0.5 \][/tex]
So, the rate of change for [tex]\( f \)[/tex] in Option D is:
[tex]\[ m = 0.5 \][/tex]
### Comparison:
We have determined the slopes of the functions as follows:
- Option B: [tex]\( m = 2 \)[/tex]
- Option C: [tex]\( m = 1 \)[/tex]
- Option D: [tex]\( m = 0.5 \)[/tex]
Among these, the highest rate of change is [tex]\( 2 \)[/tex], which corresponds to Option B.
### Conclusion:
The function [tex]\( 12x - 6y = -24 \)[/tex] in Option B is increasing at the highest rate.
