Question

Which of the following functions has the smallest $y$-intercept?

Answer 1

Let’s find the y-intercept of all the functions.

Function in option A:


Here, the slope or rate of change $=\frac{Changeinyvalues}{Changeinxvalues}=\frac{6}{2}=3$

We know the standard form of the linear function is $y=mx+b$, where m is the slope and $b$ is the $y$-intercept.
Let’s consider an input and output ordered pair of the function.
$(2,5)$ is an input and output ordered pair, where $x=2,y=5$.
Replacing the above two values of $x$ and $y$ in $y=mx+b$,

$\Rightarrow 5=3\times 2+b$
$\Rightarrow 5=6+b$
$\Rightarrow 5-6=b$
$\Rightarrow -1=b$

So, the $y$-intercept $=-1$ and that means for $x=0,y=-1$.

Function in option B:
In the table, the given input and output ordered pairs are $(0,3.5),(1,4.5),(2,5.5),$ and $(3,6.5)$.
So, the y-intercept of the function is $3.5$.

Function in option C:


Here, the slope or rate of change $=\frac{Changeinyvalues}{Changeinxvalues}=\frac{4}{2}=2$

We know that the standard form of the linear function is $y=mx+b$, where m is the slope and $b$ is the $y$-intercept.
Let’s consider an input and output order pair of the function.
$(2,5)$ is an input and output order pair, where $x=2,y=5$.
Replacing the above two values of $x$ and $y$ in $y=mx+b$,

$\Rightarrow 5=2\times 2+b$
$\Rightarrow 5=4+b$
$\Rightarrow 5-4=b$
$\Rightarrow 1=b$

So, the $y$-intercept $=1$ and that means for $x=0,y=1$.

Function in option D:

In the table given, the input and output order pairs are $(-1,-2),(0,0),(1,2),and(2,4)$.
So, the $y$-intercept of the function is $0$.

So, comparing all the four functions:
$-1<0<1<3.5$
So, the function in option A has the smallest $y$-intercept.

So, option A is the correct answer.