Problem 35 - Algebra for College Students

Question

The cost (c) of playing an online computer game for a time \((t)\) in hours is given by the equation \(c=3 t+5\). Label the horizontal axis \(t\) and the vertical axis \(c\), and graph the equation for nonnegative values of \(t\).

Step-by-Step Solution

Answer

Plot the equation \(c = 3t + 5\) starting from \(c = 5\) for \(t = 0\) on a coordinate plane; connect points to form a line.

Step 1: Understand the Equation

The given equation is a linear equation in the form of \(c = 3t + 5\), where \(c\) is the total cost, and \(t\) is the time in hours. This equation shows a direct relationship between the hours played and the cost.

Step 2: Determine Key Points

To plot the graph, identify key points by choosing values for \(t\). Start with \(t = 0\): \(c = 3(0) + 5 = 5\). For \(t = 1\): \(c = 3(1) + 5 = 8\). For \(t = 2\): \(c = 3(2) + 5 = 11\). Continue this to get enough points.

Step 3: Plot Key Points on Graph

On the graph, label the horizontal axis as \(t\) (time in hours) and the vertical axis as \(c\) (cost). Plot the calculated points: (0,5), (1,8), and (2,11). Extend this process for other values of \(t\) if needed to ensure a complete graph.

Step 4: Draw the Line

Connect the plotted points with a straight line. Since this is a linear equation, the points should lie along this line. The line represents the relationship between \(t\) and \(c\) for nonnegative values of \(t\).

Step 5: Analyze the Graph

Observe that the line intercepts the \(c\)-axis at \(c = 5\) when \(t = 0\). The slope of the line is \(3\), indicating that for each additional hour played, the cost increases by \(3\).

Key Concepts

Graphing Equations

Graphing equations is a visual method to understand how variables in equations relate to each other. For instance, in the equation \(c = 3t + 5\), we graph to see how the game cost \(c\) changes as the playtime \(t\) increases. The axes on the graph help in showing the relationship between these two quantities. The horizontal axis, or \(x\)-axis in this case, is labeled \(t\) for time. The vertical axis, or \(y\)-axis, is labeled \(c\) for cost.

• Each point on the graph corresponds to a specific value of \(t\) and \(c\).
• The graph of a linear equation will be a straight line.

Graphing equations allows you to visually see trends and relationships, such as how costs increase steadily with time in this scenario. This provides a better understanding of the situation modeled by the equation.

Slope-Intercept Form

The slope-intercept form is one of the most useful forms to write linear equations. It looks like \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. For our equation \(c = 3t + 5\), it perfectly fits into this form with \(c\) as \(y\), \(t\) as \(x\), giving:

• \(m = 3\): This slope tells us how steep the line is. Here, it means that for each hour of playtime, the cost goes up by 3 units.
• \(b = 5\): This is where the line crosses the vertical axis. It indicates the starting cost when \(t = 0\).

Using the slope-intercept form, it's easy to graph, analyze, and understand linear relationships. Knowing what each component represents helps in predicting how changes in one variable affect the other.

Plotting Points

Plotting points is a fundamental skill in graphing. It involves identifying the exact location on the graph using ordered pairs \((t, c)\). Let's take a look at how points are plotted according to our given equation.

• Start by choosing values for \(t\), such as 0, 1, 2.
• Calculate \(c\) for each value (e.g., \(t = 0\), \(c = 5\); \(t = 1\), \(c = 8\)).
• Place each ordered pair like \((0,5), (1,8), (2,11)\) on the graph accordingly.

Plotting points ensures accuracy by providing specific locations for drawing the graph. It is a step-by-step process to visualize the equation. Once all points are plotted, drawing a line through them illustrates the entire equation as a linear relationship.