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Which function is increasing at the highest rate?

A.
| [tex]\( x \)[/tex] | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| [tex]\( g(x) \)[/tex] | -5 | -4 | -3 | -2 | -1 |

B.
(Option missing)

C. [tex]\( 12x - 6y = -24 \)[/tex]

D. A linear function, [tex]\( f \)[/tex], with an [tex]\( x \)[/tex]-intercept of 8 and a [tex]\( y \)[/tex]-intercept of -4.



Answer :

GinnyAnswer
To determine which function is increasing at the highest rate, we need to analyze each option and calculate the respective rates of increase.

### Option A
The table provides the values for [tex]\( g(x) \)[/tex]:

[tex]\[ \begin{tabular}{|c|r|r|r|r|r|} \hline $x$ & 1 & 2 & 3 & 4 & 5 \\ \hline $g(x)$ & -5 & -4 & -3 & -2 & -1 \\ \hline \end{tabular} \][/tex]

We observe that [tex]\( g(x) \)[/tex] is increasing by 1 unit for each increase of 1 unit in [tex]\( x \)[/tex]. Thus, the rate of increase is:

[tex]\[ \text{Rate of increase for } g(x) = 1 \][/tex]

### Option B
No specific function or dataset is provided in Option B, making it impossible to determine a rate of increase from this option.

### Option C
We are given the equation [tex]\( 12x - 6y = -24 \)[/tex]. To find the rate of increase, we first rewrite this equation in the slope-intercept form [tex]\( y = mx + b \)[/tex].

Starting with the given equation:

[tex]\[ 12x - 6y = -24 \][/tex]

Solving for [tex]\( y \)[/tex]:

[tex]\[ -6y = -12x - 24 \][/tex]
[tex]\[ y = 2x + 4 \][/tex]

The slope [tex]\( m \)[/tex] of this linear equation is 2. Therefore, the rate of increase for this function is:

[tex]\[ \text{Rate of increase for } 12x - 6y = -24 \text{ is } 2 \][/tex]

### Option D
For this option, we are given that the linear function [tex]\( f \)[/tex] has an [tex]\( x \)[/tex]-intercept of 8 and a [tex]\( y \)[/tex]-intercept of -4. To find the rate of increase, we need to calculate the slope of the line passing through these intercepts.

The intercepts are points (8, 0) and (0, -4). The slope [tex]\( m \)[/tex] of the line can be calculated using these points:

[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-4)}{8 - 0} = \frac{4}{8} = 0.5 \][/tex]

Therefore, the rate of increase for this linear function is:

[tex]\[ \text{Rate of increase for } f(x) \text{ is } 0.5 \][/tex]

### Comparing Rates
- Option A: Rate of increase = 1
- Option B: No data given
- Option C: Rate of increase = 2
- Option D: Rate of increase = 0.5

Thus, the function with the highest rate of increase is the one given in Option C, with a rate of increase of 2.

### Conclusion
The correct answer is:

C. [tex]\( 12x - 6y = -24 \)[/tex]

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