We have the following information from the question, and we can express it in an algebraical form:
1. First Case: Sharla
[tex]2.50+3x=9.25[/tex]
2.Second Case: Deon
[tex]2.50+4x=11.50[/tex]
If we solve the value for both equations, we get that the value for x is constant, and it is x = 2.25. This is the value of the slope of this line equation:
[tex]3x=9.25-2.50\Rightarrow3x=6.75\Rightarrow x=2.25[/tex]
If we solve for x in the second equation (Deon) we get the same result.
(m = 2..25)
[tex]f(t)=2.50+2.25t[/tex]
Now we have the line equation for this situation.
a. The Rate of Change
The rate of change is given by the formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
And we have that
Sharla:
x1 = 3 miles
y1 = $9.25
Deon:
x2 = 4 miles
y2 = $11.50
Then, we have:
[tex]m=\frac{11.50-9.25}{4-3}\Rightarrow m=2.25[/tex]
As we can see, we got this value earlier, and it is the rate of change for this linear function, and it is also known as the slope of the line.
In summary, the rate of change is 2.25 dollars per mile.
b. Initial Value for the Cost
The initial value for the cost is $2.50 (base fee). If we see the function for the cost below, if we have that before travelling, Sharla and Deon have to pay $2.50, and it represents the base fee in this context.
For 0 miles, we have that the function gives us:
[tex]y=2.50+2.25\cdot0\Rightarrow y=2.50[/tex]
c. The Function that represents the cost of a cab ride, y, in terms of miles traveled, x is:
[tex]y=2.50+2.25x[/tex]
This function represents costs. If we check this for Sharla, we have that she travels for three miles, then, we have:
[tex]y=2.50+2.25\cdot3=2.50+6.75\Rightarrow y=9.25[/tex]
Sharla spent $9.25 for 3 miles. The same applies to Deon.
Therefore, the function is:
[tex]y=2.50+2.25x[/tex]
