Use the equation for delivery costs found in Exercise 28 to...

Use the equation for delivery costs found in Exercise 28 to answer the following: a. Give the slope of the equation and explain its meaning in this situation. b. If the fees were changed to $\$ 7$ for delivery and 75 cents for each mile over 10 miles, how will this change the appearance of the graph?

Use the equation for delivery costs found in Exercise 28 to answer the following:
a. Give the slope of the equation and explain its meaning in this situation.
b. If the fees were changed to $\$ 7$ for delivery and 75 cents for each mile over 10 miles, how will this change the appearance of the graph?

Key Concepts

Graph Transformations

Graph transformations involve shifting, stretching, or otherwise modifying the graph of a function as a result of changing its equation's parameters. Alterations in the fixed fee or the per-unit rate in a cost equation, for example, will change the y-intercept and the slope, respectively, thereby affecting the line's position and steepness. Recognizing these transformations helps in visualizing and interpreting how modifications to a model impact its graphical representation.

Y-Intercept

The y-intercept of a linear equation represents the value of the dependent variable when the independent variable is zero. In the context of a cost function, it often corresponds to the base fee or fixed cost that is incurred regardless of additional variable amounts (such as distance). This concept is crucial when distinguishing between fixed and variable components of a model.

Linear Equations

Linear equations provide a way to model relationships between two variables with a constant rate of change. In many real-world situations, such as calculating delivery costs, a linear equation can represent a fixed starting cost (the y-intercept) plus an additional cost that increases linearly with distance (captured by the slope). Understanding linear equations is essential for interpreting and predicting such relationships.

Slope

The slope in a linear function represents the rate of change of the dependent variable with respect to the independent variable. In practical applications like cost functions, it tells you how much the output (such as cost) will change for each additional unit of the input (such as mileage). This concept is fundamental in understanding how changes in one quantity affect another in a consistent, linear manner.

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