Skate and Climb: Straight-Line Rule Checks

Decide which gym rules describe linear functions and which do not.

Membership Rules

Read each gym rule. A linear function has a straight-line graph and can be written as y = mx + b.

A gym membership costs $18 per month plus $4 per visit. The rule y = 4x + 18 gives the total monthly cost for x visits. Which statement describes its graph?

A. It is a straight line that passes through (0, 18).B. It is a straight line that passes through (0, 4).C. It is a curve that passes through (18, 0).D. It is a curve that passes through (0, 18).

A prepaid skate pass has a remaining balance of y = 42 − 6x dollars after x sessions. Rewrite the rule mentally as y = mx + b.

What is the value of m? What is the value of b?

An unlimited monthly membership costs $37, regardless of the number of visits. Its rule is y = 37. How should this function be classified?

A. It is nonlinear because its rule does not display an x-term.B. It is nonlinear because the total cost never increases.C. It is linear because y = 37x has a rising straight-line graph.D. It is linear because y = 0x + 37 has a horizontal straight-line graph.

The gym is testing these function rules. Classify each rule as linear or nonlinear.

A climbing pass costs $7 per visit for the first 3 visits. After that, the monthly total stays at $21. Consider the total cost as a function of the number of visits, using whole-number inputs. Which statement is correct?

A. It is linear because each visit always adds $7 to the total.B. It is nonlinear because the rate changes from $7 to $0 per visit.C. It is nonlinear because the first three totals increase.D. It is linear because each visit has exactly one total cost.

Route Records and Graphs

Use the tables and plotted points. Compare changes in y for equal changes in x, not just changes between neighboring rows.

A route-design tool records height y at horizontal position x. Which statement describes the data?

Route-Design Points
x (feet)	y (feet)
0	2
1	5
3	11
5	17

A. The points fit a nonlinear rule because the y-changes are 3, 6, and 6.B. The points fit a linear rule because y increases 3 feet per foot of x.C. The points fit a nonlinear rule because the first y-value is not zero.D. The points fit a linear rule because y increases 6 feet per foot of x.

A straight practice ramp follows a linear height function. Use the table to find its missing height.

Straight Practice Ramp
Distance x (feet)	Height y (feet)
0	1
2	2
4	3
8	?

At x = 8 feet, the height is feet.

A route-design screen plots a function at five inputs. Could all five points belong to one linear function?

A. No, because the five points do not lie on one straight line.B. Yes, because one plotted point is at the origin.C. Yes, because each plotted input has exactly one output.D. No, because linear functions cannot have negative inputs.

A climbing challenge awards y points for a route with difficulty level x. Which rule fits every row, giving an example of a nonlinear function?

Challenge Points
Difficulty x	Points y
1	2
2	5
3	10
4	17

A. y = 3x − 1B. y = 2xC. y = x² + 1D. y = x + 1

10.

Select every rule whose entire graph is a straight line. Unless a restriction is shown, use all real inputs.

Design Decisions

Apply what you know about straight-line graphs and nonlinear functions.

11.

The gym marks a square landing zone with side length x meters. Its area is y square meters. Which rule and classification are correct?

A. y = x²; the function is nonlinear.B. y = x²; the function is linear.C. y = 4x; the function is linear.D. y = 2x; the function is linear.

12.

A 30-minute climbing session is shared equally among x climbers. Each climber gets y = 30/x minutes, where x is a positive whole number. Why is this function nonlinear?

A. Every rule with a fraction has a curved graph.B. Every rule with only positive inputs is nonlinear.C. Increasing x by 1 does not produce a constant change in y.D. Each allowed x produces more than one possible value of y.

13.

Noor says the route rule y = x(x + 2) − x² is nonlinear because it contains x². Which response correctly checks the rule?

A. It simplifies to y = 2, so the graph is a horizontal line.B. It simplifies to y = x² + 2, so the graph is not a straight line.C. It simplifies to y = 2x², so the graph is not a straight line.D. It simplifies to y = 2x, so the graph is a straight line.

14.

Create a nonlinear function for the points y earned on a climbing route with difficulty level x. Use positive whole-number difficulty levels. Give your rule and its outputs for x = 1, 2, and 3. Explain how those outputs show that your function is not linear.