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 growth 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-and-drink sales in $2015 .$ d. From parts (a), (b), and (c), determine whether the restaurant food-and-drink sales are 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 growth 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-and-drink sales in $2015 .$
d. From parts (a), (b), and (c), determine whether the restaurant food-and-drink sales are increasing at a steady rate. Explain why or why not.

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Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial function of the form ax² + bx + c. It is used to model relationships where the rate of change is not constant, and its graph is a parabola. This concept is crucial when analyzing growth patterns that accelerate over time rather than increase at a linear pace.

Function Evaluation

Function evaluation involves substituting specific input values into a function to calculate the corresponding output values. This process is fundamental in applying mathematical models to real-world scenarios to estimate outcomes for given independent variable values.

Variable Transformation

Variable transformation is the technique of converting real-world measurements into the variables used in a mathematical model. In this context, it involves relating calendar years to the variable in the function (for example, using the number of years since a base year), which is essential for accurate application of the function.

Rate of Change Analysis

Rate of change analysis examines how the output of a function changes with respect to changes in its input. For a quadratic function, the rate of change is not constant but increases or decreases over time, which is an important consideration when determining whether growth occurs steadily or at an accelerating rate.

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