Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
