The graph of the function f, where \(\mathrm{y = f(x)}\), gives the total cost y, in dollars, for a certain...

1. TRANSLATE the problem information

Looking at the graph and problem statement:

• Given information:
  • y = total cost in dollars for system and games
  • x = number of games
  • The graph shows a linear relationship
  • We need to interpret what the slope means

• What we need to find: The meaning of the slope in this real-world context

2. TRANSLATE coordinates from the graph

To find slope, we need two clear points on the line:

• Point 1: (0, 100)
  • When there are 0 games, the total cost is $100
  • This makes sense—it's the cost of just the video game system

• Point 2: (1, 125)
  • When there is 1 game, the total cost is $125

3. SIMPLIFY to calculate the slope

Using the slope formula:

\(\mathrm{Slope} = \frac{\mathrm{change~in~y}}{\mathrm{change~in~x}} = \frac{\mathrm{y_2 - y_1}}{\mathrm{x_2 - x_1}}\)

\(\mathrm{Slope} = \frac{125 - 100}{1 - 0}\)

\(\mathrm{Slope} = \frac{25}{1}\)

\(\mathrm{Slope} = 25\)

4. INFER what the slope means in context

Here's the key insight:

• \(\mathrm{Slope} = \frac{\mathrm{change~in~total~cost}}{\mathrm{change~in~number~of~games}}\)
• This tells us: how much the total cost increases for each additional game
• In other words: the cost per game

Since the slope is 25, each game costs $25.

Check against answer choices:

• Choice A: 'Each game costs $25' ✓ This matches our interpretation
• Choice B: 'The video game system costs $100' - This is the y-intercept, not the slope
• Choice C: 'The video game system costs $25' - Wrong value for the system
• Choice D: 'Each game costs $100' - Wrong value for the slope

Answer: A. Each game costs $25.

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion between slope and y-intercept:

Students often mix up what slope and y-intercept represent. They see that the line starts at 100 on the y-axis and think 'the question asks about the slope, and I see 100 on the graph, so the slope must relate to $100.'

This confusion leads them to select Choice B (The video game system costs $100) or Choice D (Each game costs $100), interpreting features of the graph incorrectly.

Second Most Common Error:

Weak INFER skill—not connecting slope to rate of change:

Some students can calculate the slope correctly (getting 25) but don't understand what it means in context. They might think:

• '25 is smaller than 100, so 25 must be some minor cost'
• 'The system is the main purchase, so slope must relate to the system'

This leads them to select Choice C (The video game system costs $25), applying the correct calculation to the wrong interpretation.

The Bottom Line:

The key challenge is understanding that slope always represents rate of change. In this context, slope = (change in total cost)/(change in games) = cost per game. The y-intercept (where the line crosses the y-axis when x = 0) represents the initial fixed cost—the video game system itself. Many students confuse these two fundamental features of linear graphs.