The function f(x)=0.19 x^2+5.67 x+43.7 can be used to...

The function $f(x)=0.19 x^{2}+5.67 x+43.7$ can be used to approximate the amount of restaurant food-and-drink sales, where $x$ is the number of years since 1970 and $f(x)$ or $y$ is the sales (in billions of dollars.) a. Approximate the restaurant food-and-drink sales in 2005 b. Approximate the restaurant food-and-drink sales in 2010 c. Use this function to estimate the restaurant food-anddrink sales in 2015 d. From parts $(a),(b),$ and $(c),$ determine whether the restaurant food-and-drink sales is increasing at a steady rate. Explain why or why not.

The function $f(x)=0.19 x^{2}+5.67 x+43.7$ can be used to approximate the amount of restaurant food-and-drink sales, where $x$ is the number of years since 1970 and $f(x)$ or $y$ is the sales (in billions of dollars.)
a. Approximate the restaurant food-and-drink sales in 2005
b. Approximate the restaurant food-and-drink sales in 2010
c. Use this function to estimate the restaurant food-anddrink sales in 2015
d. From parts $(a),(b),$ and $(c),$ determine whether the restaurant food-and-drink sales is increasing at a steady rate. Explain why or why not.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function of degree two, generally written in the form ax² + bx + c. This type of function often models situations where the change is not constant, reflecting phenomena that accelerate or decelerate over time. In applications involving prediction over time, such as forecasting sales, the quadratic function captures changes that may become more or less pronounced as time progresses.

Function Evaluation

Function evaluation involves substituting a specific value of the independent variable into a function to compute the corresponding output. In the context of prediction, one uses the function by plugging in the given time variables (after any necessary transformation) to estimate the corresponding quantity, such as restaurant sales for a particular year.

Independent Variable Transformation

An independent variable transformation is the process of reassigning the variable to better suit the context of a problem. In many time-based models, the year is transformed into a variable representing the number of years since a specific starting point, such as 1970. This standardizes the variable and simplifies the function, making computations and interpretations more straightforward.

Rate of Change

Rate of change in a function is a measure of how the output value changes as the input changes. For functions modeled by a quadratic equation, the rate of change is not constant; rather, it varies with the independent variable. Comparing differences in function outputs over equal intervals can indicate whether the underlying process is accelerating, decelerating, or increasing at a steady rate, which is crucial for understanding trends and making future predictions.

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