Question 678·Easy·Algebra

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The table shows several values of $n$ and their corresponding values of $p$ .

n	p
0	-4
1	2
3	14

There is a linear relationship between $n$ and $p$ . Which equation represents this relationship?

A table described as a linear relationship calls for two facts: how much the output changes per input and what the output is at input $0$ . Enter the pairs in Desmos, then fit $y_1\sim mx_1+b$ to find both. Use the table’s variable names in the final rule. Watch the sign of the zero-input value; its size alone isn’t enough.

Hints

Hint 1
Each table row is a point $(n,p)$ : the first number is the input, and the second is its output. Enter the pairs in a Desmos table. Which row shows the output at input $0$ ?
Hint 2
In $p=mn+b$ , the slope $m$ is how much $p$ changes when $n$ increases by $1$ . Fit $y_1\sim mx_1+b$ to the table in Desmos. Which value under PARAMETERS gives that change per input?

Step-by-step

Fit a linear rule to the table
Step 1
Enter the table pairs
Enter the table pairs in Desmos, using $x_1$ for $n$ and $y_1$ for $p$ . Desmos plots $(0,-4)$ , $(1,2)$ , and $(3,14)$ . Each point pairs an input with its output, so $(0,-4)$ means $p=-4$ when $n=0$ .
Step 2
Use the zero-input row
Write a linear rule as $p=mn+b$ . Here $m$ is the change in $p$ for each increase of $1$ in $n$ , and $b$ is $p$ when $n=0$ . The zero-input row gives $b=-4$ , so $p=mn-4$ .
Step 3
Find the slope in Desmos
Keep the table and type $y_1\sim mx_1+b$ . The $\sim$ tells Desmos to fit the table points. Under PARAMETERS, it shows $m=6$ and $b=-4$ , so $p$ rises by $6$ for each extra $n$ . One row fixes where a line starts; its slope fixes how it changes.
Step 4
Write the equation
Put $m=6$ into $p=mn-4$ :
$p=6n-4.$
The negative constant agrees with $p=-4$ at $n=0$ ; $+4$ would not. So this equation represents the relationship between $n$ and $p$ . Choice C.

Question 678·Easy·Algebra

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