Question

$M$ is a function whose graph includes the points $(-3,-3),(-1,1),(0,3),\text{and}(4,11)$.
$N$ is a function represented by the table shown below.

$x$	$y$
$-1$	$-9$
$0$	$-5$
$1$	$-1$
$2$	$3$

Which of the following options is correct?

• A. Both functions are linear.
• B. Function $M$ is linear, but function $N$ is non-linear.
• C. Function $N$ is linear, but function $M$ is non-linear.
• D. Both functions are non-linear.

Answer 1

The correct option is A Both functions are linear.

We know that, in a linear function, the slope is constant for any two points on the line.

Function $M$

Given, the points included in the graph of function $M$ are
$(-3,-3),(-1,1),(0,3),\text{and}(4,11)$.

The slope between the first and the second points $=\frac{(1-(-3\left)\right)}{(-1-(-3\left)\right)}=\frac{(1+3)}{(-1+3)}=\frac{4}{2}=2$

The slope between the second and the third points $=\frac{(3-1)}{(0-(-1\left)\right)}=\frac{2}{1}=2$

The slope between the third and the fourth points $=\frac{(11-3)}{(4-0)}=\frac{8}{4}=2$

Since the slope is constant, the given function will be linear.

Function N

Given:

$x$	$y$
$-1$	$-9$
$0$	$-5$
$1$	$-1$
$2$	$3$

The slope between the first and the second points $=\frac{(-5-(-9\left)\right)}{(0-(-1\left)\right)}=\frac{(-5+9)}{(0+1)}=\frac{4}{1}=4$

The slope between the second and the third points $=\frac{(-1-(-5\left)\right)}{(1-0)}=\frac{(-1+5)}{1}=\frac{4}{1}=4$

The slope between the third and the fourth points $=\frac{(3-(-1\left)\right)}{(2-1)}=\frac{(3+1)}{1}=\frac{4}{1}=4$

Since the slope is constant, the given function will be linear.

Hence, both the functions $M$ and $N$ are linear.

Option C is correct.