Question

Which of the following represents a function which is linear and proportional?

• A.

  $x$	$0.5$	$4.5$	$6.5$	$7.5$
  $y$	$4$	$36$	$52$	$60$

• B.

  $x$	$0.25$	$0.75$	$1.75$	$2.5$
  $y$	$-7$	$-3$	$5$	$11$

• C.

  $x$	$-1$	$2$	$5$	$9$
  $y$	$-5$	$16$	$37$	$65$

• D.

  $x$	$-1$	$2$	$3$	$5$
  $y$	$-3$	$0$	$5$	$21$

Answer 1

The correct option is B

$x$	$0.25$	$0.75$	$1.75$	$2.5$
$y$	$-7$	$-3$	$5$	$11$

Let’s find the slope between the points to check whether the function is linear or not.

Option A

$x$	$0.25$	$0.75$	$1.75$	$2.5$
$y$	$-7$	$-3$	$5$	$11$

The slope between the first and the second points $=\frac{\left(\right(-3)-(-7\left)\right)}{(0.75-0.25)}=\frac{(-3+7)}{0.5}=\frac{4}{0.5}=8$

The slope between the second and the third points $\frac{(5-(-3\left)\right)}{(1.75-0.75)}=\frac{(5+3)}{1}=\frac{8}{1}=8$

The slope between the third and the fourth points $=\frac{(11-5)}{(2.5-1.75)}=\frac{6}{0.75}=8$

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

Now, the slope between the origin $(0,0)$ and the first point $=\frac{\left(\right(-7)-0)}{(0.25-0)}=\frac{-7}{0.75}=-28$

Since the slope between the first point and the origin is not the same as the slope between the other pair of points, we can say that the graph of the given function will not pass through the origin.

We know that only the linear functions that pass through the origin are proportional.
Hence, the given function is not proportional.

Option B

$x$	$1$	$2$	$5$	$9$
$y$	$-5$	$16$	$37$	$65$

The slope between the first and the second points $=\frac{(16-(-5\left)\right)}{(2-(-1)}=\frac{(16+5)}{(2+1)}=\frac{21}{3}=7$

The slope between the second and the third points $\frac{(37-16)}{(5-2)}=\frac{21}{3}=7$

The slope between the third and the fourth points $=\frac{(65-37)}{(9-5)}=\frac{28}{4}=7$

Now, the slope between the origin $(0,0)$ and the first point $\frac{\left(\right(-5)-0)}{(-1)-0}=\frac{-5}{-1}=5$

Since the slope between the first point and the origin is not the same as the slope between the other pair of points, we can say that the graph of the given function will not pass through the origin.

We know that only the linear functions that pass through the origin are proportional.
Hence, the given function is not proportional.

Option C

$x$	$-1$	$2$	$3$	$5$
$y$	$-3$	$0$	$5$	$21$

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

The slope between the second and the third points $\frac{(5-0)}{(3-2)}=\frac{5}{1}=5$

The slope between the third and the fourth points $=\frac{(21-5)}{(5-3)}=\frac{16}{2}=8$

Since the rate is not constant, the given function is not linear.

Option D

$x$	$0.5$	$4.5$	$6.5$	$7.5$
$y$	$4$	$36$	$52$	$60$

The slope between the first and the second points $=\frac{(36-4)}{(4.5-0.5)}=\frac{32}{4}=8$

The slope between the second and the third points $\frac{(52-36)}{(6.5-4.5)}=\frac{16}{2}=8$

The slope between the third and the fourth points $=\frac{(60-52)}{(7.5-6.5)}=\frac{8}{1}=8$

Now, the slope between the origin $(0,0)$ and the first point $=\frac{(4-0)}{(0.5-0)}=\frac{4}{0.5}=8$

Since the slope between the first point and the origin is the same as the slope between the other pair of points, we can say that the graph of the given function will pass through the origin.

We know that only the linear functions that pass through origin are proportional.
Hence, the given function is proportional.

Option D is correct.