Question

can i determine rate of change and initial values from tables and graphs
(partially visible problem about t - shirts and total cost table, with sub - questions a, b, c about equation, slope meaning, y - intercept meaning)
i can compare properties of functions.

1. in new york city, the cost to take a taxi can be represented with the equation $y = 1.40x+2.50$, where $x$ is the number of miles driven and $y$ is the total cost. the cost of a taxi in san francisco is shown in the table.

(table with miles: 0, 5, 10, 15, 20 and total cost: $2.65, $5.10, $7.55, $9.00, $11.65)
a. in which city is the rate per mile higher?
b. in which city is the initial fee higher?

1. jill and carly each buy a box of gummy candy at the movies. the number of gummies carly has remaining, $y$, based on the number of minutes spent eating, $x$, can be represented by the equation $y=-0.2x + 22$. the number of gummies jill has left based on the amount of time is shown in the graph below.

(graph with x - axis time (minutes) and y - axis gummies remaining)
a. who started with the most gummies in their box?
b. who is eating their gummies at a faster rate?

Explanation:

First Problem (T-shirts)

Step1: Find rate of change

Rate = $\frac{12.24 - 5.94}{1 - 0} = 6.30$

Step2: Identify initial value

Initial cost (0 shirts) = $\$5.94$

Step3: Write linear equation

$y = 6.30x + 5.94$

Step4: Interpret slope

Slope = cost per t-shirt: $\$6.30$

Step5: Interpret y-intercept

Y-intercept = base cost: $\$5.94$

Step1: Find NYC rate & initial fee

NYC equation: $y=0.40x+2.50$, so rate = $\$0.40$/mile, initial fee = $\$2.50$

Step2: Find SF rate of change

SF rate = $\frac{5.10 - 2.60}{5 - 0} = 0.50$

Step3: Find SF initial fee

SF initial fee (0 miles) = $\$2.60$

Step4: Compare rates

$0.50 > 0.40$, so SF rate is higher

Step5: Compare initial fees

$2.60 > 2.50$, so SF initial fee is higher

Step1: Find Carly's initial gummies

Carly's equation: $y=-0.2x+22$, initial (x=0) = 22 gummies

Step2: Find Jill's initial gummies

Jill's graph starts at (0,20), so initial = 20 gummies

Step3: Compare initial amounts

$22 > 20$, so Carly has more initially

Step4: Find Carly's eating rate

Carly's rate = $|-0.2| = 0.2$ gummies per minute

Step5: Find Jill's eating rate

Jill's rate = $\frac{20 - 14}{15 - 0} = 0.4$ gummies per minute

Step6: Compare eating rates

$0.4 > 0.2$, so Jill eats faster

Answer:

a. $y = 6.30x + 5.94$
b. The slope means each t-shirt costs $\$6.30$.
c. The y-intercept means there is a $\$5.94$ base cost (when 0 t-shirts are purchased).

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Second Problem (Taxi Costs)