Question

Which of the following functions have the same rate of change and the same y-intercept as $y=5x-\frac{7}{2}$?

Answer 1

The given function is $y=5x-\frac{7}{2}$.
By comparing this function with the standard form of a linear function $y=mx+b$, we get:
$m=5,b=-\frac{7}{2}$
That means the slope or rate of change $=5$,
and the $y$-intercept $=-\frac{7}{2}=-3.5$.

Now, we will identify the slope and $y$-intercept of all the functions given in the options.

Function $1$: A photographer charges a fixed price of $\$5$ to shoot a video of $5$ minutes and a variable charge of $\$3.5$ for each additional minute.
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, the slope $=3.5$ and $y$-intercept $=5$.

So, the slope or rate of change will be $=\frac{Changeinyvalues}{Changeinxvalues}=\frac{2-(-5)}{2-0}$

$=\frac{2+5}{2}=\frac{7}{2}=3.5$

And the straight line touches the $y$-axis at $(0,-5)$. So, the $y$-intercept is $-5$.

Function 2: Let’s consider two points from the graph passing through the straight line.
$(0,-5)$ and $(2,2)$ are two points.

Function 3:


Here, the slope or rate of change will be $=\frac{Changeinyvalues}{Changeinxvalues}=\frac{5}{1}=5$

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 order pair of the function.
$(1,1.5)$ is an input and output order pair, where $x=1,y=1.5$.

Replacing the above two values of x and y in $y=mx+b$,
$\Rightarrow 1.5=5\times 1+b$
$\Rightarrow 1.5=5+b$
$\Rightarrow 1.5-5=b$
$\Rightarrow -3.5=b$

So, the $y$-intercept $=-3.5$ and that means for $x=0,y=-3.5$.

Now, comparing the slope and $y$-intercept of all the three functions with the given function, we can observe that function $3$ has the same slope and $y$-intercept as $y=5x-\frac{7}{2}$.

So, option C is the correct answer.