Question 697·Medium·Algebra

0 / 700

The ordered pairs $(2,5)$ , $(6,17)$ , and $(14,41)$ all lie on the graph of $g$ , a linear function defined by $g(x)=px+q$ , where $p$ and $q$ are constants.

What is the value of $q$ ?

A linear function changes by a fixed amount for each $1$ increase in its input; that amount is its slope. With two points, divide the change in output by the change in input, then use both coordinates from one point to find the constant in the rule. Desmos can calculate the slope and fit the remaining unknown. An output given at an input other than $0$ is not the constant.

Hints

Hint 1
The slope is the change in output divided by the change in input. Subtract the coordinates of two points in the same order on top and bottom. What slope do the first two pairs give you?
Hint 2
An ordered pair $(x,y)$ on a function means input $x$ produces output $y$ . Once you know $p$ , put both coordinates from $(2,5)$ into $g(x)=px+q$ . Which letter remains unknown?

Step-by-step

Find the slope, then fit the constant
Step 1
Find the slope from two points
The slope $p$ is the change in output divided by the change in input. Using $(2,5)$ and $(6,17)$ , type $\frac{17-5}{6-2}$ in Desmos. It prints $3$ , so $p=3$ . Subtract in the same order on top and bottom.
Step 2
Turn one point into an equation
The point $(2,5)$ means $g(2)=5$ . Put its input and output, along with $p=3$ , into $g(x)=px+q$ : $5=3(2)+q$ . This uses both coordinates from the same point. The output $5$ is not $q$ : it occurs at $x=2$ , not $x=0$ .
Step 3
Solve for the constant
Type $5\sim3(2)+q$ in Desmos. The tilde $\sim$ tells it to fit the unknown $q$ ; under PARAMETERS, Desmos reports $q=-1$ . At $x=0$ , the $px$ term is $0$ , so $q$ is the function's output at input $0$ . The value of $q$ is $-1$ . Choice B.

Question 697·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this