Question

Which of the following two functions have the same rate of change and the same y-intercept?
Function $1$: $y=4.5x-3$
Function $2$: An event management company charges a fixed price of $\$3$ to book a wedding sofa and a variable charge of $\$4.5$ for each day it will be booked.
Function $3$:


Function $4$:

$x$	$0$	$2$	$4$	$6$
$y$	$3.5$	$-2.5$	$-8.5$	$-14.5$

• A. Function $1$ and Function $3$
• B. Function $1$ and Function $2$
• C. Function $2$ and Function $4$
• D. Function $3$ and Function $4$

Answer 1

The correct option is A Function $1$ and Function $3$

Let’s identify the slope and y-intercept of all the four functions.

Function $1$: $y=4.5x-3$
Comparing Function $Z$ with the standard form of a linear function $y=mx+b$.
$m=4.5,b=-3$
So, the slope or rate of change $=4.5$ and $y$-intercept $=-3$.

Function $2$: An event management company charges a fixed price of $\$3$ to book a wedding sofa and a variable charge of $\$4.5$ for each day it will be booked.
The fixed rate is always considered as the $y$-intercept,
and the variable rate is considered as the rate of change or slope.
So, here slope $=4.5$ and $y$-intercept $=3$.

Function 3: Let’s consider two points from the graph passing through the straight line.
$(0,-3)$ and $(2,6)$ are two points.

So, the slope or rate of change $=\frac{Changeinyvalues}{Changeinxvalues}=\frac{6-(-3)}{2-0}$

$=\frac{6+3}{2}=\frac{9}{2}=4.5$

And the straight line touches the $y$-axis at $(0,-3)$. So, the $y$-intercept is $-3$.

Function $4$:


Here, the slope or rate of change $=\frac{Changeinyvalues}{Changeinxvalues}=-\frac{6}{2}=-3$
And for $0$ as input, the output is $3.5$.
So, $y$-intercept $=3.5$.

Now on comparing the slope and $y$-intercept of all the functions,
we can observe that slope and $y$-intercept of Function $1$ and Function $3$ are the same.

So, option B is the correct answer.