Answer:
Option B: [tex]12x - 6y = -24[/tex]
Step-by-step explanation:
A function is said to be increasing if it has a slope greater than 0.
The function with the largest slope has the highest increasing rate.
Here, for the graph in option A, the function is decreasing with increasing [tex]x[/tex]. So, it has a negative slope.
We need to convert equation of option B into standard form and then determine its slope.
The standard form of a straight line is [tex]y = mx + b[/tex], where [tex]m[/tex] is the slope of the line and [tex]b[/tex] is the y-intercept.
Now, given function is:
[tex]12x - 6y = -24\\ 6y = 12x + 24\\ y = \frac{12}{6}x+\frac{24}{6}\\ y = 2x + 4[/tex]
Therefore, the slope is 2.
Option C:
Slope is given as,
[tex]m_{C}=\frac{g(x_{2}-g(x_{1})}{x_{2}-x_{1}} =\frac{-4-(-5)}{2-1}=\frac{-4+5}{1}=1[/tex]
Therefore, the slope is 1.
Option D:
Equation of a line with [tex]a[/tex] and [tex]b[/tex] as [tex]x[/tex] and [tex]y[/tex] intercepts is given as,
[tex]\frac{x}{a}+\frac{y}{b}=1[/tex]
Here, [tex]a = 8[/tex] and [tex]b = -4[/tex]
∴ [tex]\frac{x}{8}+\frac{y}{-4}=1\\ \frac{x-2y}{8}=1\\ x-2y=8\\ 2y=x-8\\y=\frac{1}{2}x-4[/tex]
Therefore, the slope is [tex]\frac{1}{2}[/tex].
Hence, the largest slope is for option B and thereby the highest increasing rate.
