Question

question 3: determine which representation – table of values, graph, or equation – you would rather use to find the x-intercept of the function that represents the total cost of a cell phone plan with an enrollment fee and a monthly fee after a certain number of months. justify your choice.

Explanation:

To find the \( x \)-intercept (where total cost is $0$) for the cell - phone plan cost function (which is a linear function, \( C(x)=mx + b \), with \( m \) as monthly fee and \( b \) as enrollment fee), using the equation is efficient.

1. For a table of values: We would need to generate a table with different \( x \) (months) and \( C(x) \) (cost) values. To find when \( C(x) = 0 \), we may have to do a lot of calculations or interpolate if the \( x \)-intercept is not at one of the table's \( x \)-values.
2. For a graph: We would need to plot the points (either from a table or from the equation) and then estimate the \( x \)-intercept from the graph. Estimation can lead to inaccuracies, especially if we need a precise value.
3. For the equation: The cost function for the cell - phone plan is linear, \( C(x)=mx + b \), where \( m>0 \) (monthly fee) and \( b>0 \) (enrollment fee). To find the \( x \)-intercept, we set \( C(x) = 0 \), so \( 0=mx + b \). Then we solve for \( x \) using algebraic manipulation: \( mx=-b\), and \( x =-\frac{b}{m}\). This gives us an exact value (even though in the context of months, we might interpret it as a theoretical point where the cost is zero) without the need for estimation or extensive table - building.

Answer:

I would rather use the equation. The cost function for the cell - phone plan is a linear function of the form \( C(x)=mx + b \) (where \( m \) is the monthly fee and \( b \) is the enrollment fee). To find the \( x \)-intercept, we set \( C(x) = 0 \) (since the \( x \)-intercept is where the total cost is $0$). Solving \( 0=mx + b \) for \( x \) gives \( x=-\frac{b}{m} \), which is an exact calculation. Using a table of values would require either extensive calculation or interpolation, and using a graph would involve estimation, both of which are less precise than directly solving the linear equation.