Question

Which function is increasing at the highest rate?

Answer 1

Answer:

Option B: $12x - 6y = -24$

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 $x$. 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 $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept.

Now, given function is:

$12x - 6y = -24\\ 6y = 12x + 24\\ y = \frac{12}{6}x+\frac{24}{6}\\ y = 2x + 4$

Therefore, the slope is 2.

Option C:

Slope is given as,

$m_{C}=\frac{g(x_{2}-g(x_{1})}{x_{2}-x_{1}} =\frac{-4-(-5)}{2-1}=\frac{-4+5}{1}=1$

Therefore, the slope is 1.

Option D:

Equation of a line with $a$ and $b$ as $x$ and $y$ intercepts is given as,

$\frac{x}{a}+\frac{y}{b}=1$

Here, $a = 8$ and $b = -4$

∴ $\frac{x}{8}+\frac{y}{-4}=1\\ \frac{x-2y}{8}=1\\ x-2y=8\\ 2y=x-8\\y=\frac{1}{2}x-4$

Therefore, the slope is $\frac{1}{2}$.

Hence, the largest slope is for option B and thereby the highest increasing rate.