Notice that the shape of the temperature graph for Exercises...

Notice that the shape of the temperature graph for Exercises 81 through 84 is similar to a parabola (see Section 8.5). In fact, this graph can be approximated by the quadratic function $f(x)=3 x^{2}-18 x+57,$ where $f(x)$ is the temperature in degrees Fahrenheit and $x$ is the number of days from Sunday. Use this function to answer. Use the given function and the quadratic formula to find when the low temperature was $35^{\circ} \mathrm{F}$. [Hint: Let $f(x)=35$ and solve for $x .]$ Round your answer to one decimal place and interpret your result. Does your answer agree with the graph on the previous page?

Notice that the shape of the temperature graph for Exercises 81 through 84 is similar to a parabola (see Section 8.5). In fact, this graph can be approximated by the quadratic function $f(x)=3 x^{2}-18 x+57,$ where $f(x)$ is the temperature in degrees Fahrenheit and $x$ is the number of days from Sunday. Use this function to answer.
Use the given function and the quadratic formula to find when the low temperature was $35^{\circ} \mathrm{F}$. [Hint: Let $f(x)=35$ and solve for $x .]$ Round your answer to one decimal place and interpret your result. Does your answer agree with the graph on the previous page?

Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial function of the form ax² + bx + c, where a, b, and c are constants. It produces a graph in the shape of a parabola, which may open upwards or downwards depending on the coefficient a. Understanding the properties of quadratic functions is important for modeling and analyzing real-world phenomena that change non-linearly.

Quadratic Equations

A quadratic equation is an equation that equates a quadratic function to a given value, and solving it means finding the values of the variable that satisfy the equation. This concept is essential in various applications, as it allows us to determine key points such as maximum, minimum, or specific values of interest in the modeled situation.

Quadratic Formula

The quadratic formula, x = (–b ± ?(b² – 4ac)) / (2a), is a universal method used to solve quadratic equations. It is particularly useful when factoring is difficult or impossible, providing a systematic way to find the variable's values that make the quadratic equation true. This formula relies on the discriminant, b² – 4ac, to determine the nature and number of solutions.

Graph Interpretation

Interpreting the graph of a quadratic function involves analyzing the shape and key features of the parabola, such as its vertex, axis of symmetry, and intercepts. This ability allows one to relate algebraic solutions to real-world phenomena; for example, determining when a modeled temperature reaches a specific value by observing the corresponding point on the parabola.

Rounding and Approximation

Rounding and approximation are practical mathematical techniques used to simplify numbers to a desired degree of accuracy. When solving quadratic equations for real-world applications, rounding ensures that the results are reasonable and understandable, acknowledging that exact values may not be necessary or physically meaningful in context.

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