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Chapter 8: Problem 77

Without calculating, tell whether each graph has a minimum value or a maximum value. See the Concept Check in the section. $$ f(x)=3-\frac{1}{2} x^{2} $$

Short Answer

Expert verified
The graph has a maximum value.

Step by step solution

01

Identify the Form of the Function

The given function is a quadratic function of the form \( f(x) = ax^2 + bx + c \). In this exercise, the function is \( f(x) = 3 - \frac{1}{2}x^2 \), which can be rewritten as \( f(x) = -\frac{1}{2}x^2 + 0x + 3 \).
02

Determine the Coefficient of \( x^2 \)

In the quadratic function \( ax^2 + bx + c \), the coefficient \( a \) is \(-\frac{1}{2}\) for the given function. This tells us that \( a \) is negative.
03

Analyze the Sign of the Coefficient \( a \)

For quadratic functions, if \( a > 0 \), the parabola opens upwards and has a minimum value. If \( a < 0 \), the parabola opens downwards and has a maximum value. Because \( a = -\frac{1}{2} \) and it is negative, the parabola will open downwards.
04

Conclude the Type of Value

Since the parabola opens downwards, the graph has a maximum value rather than a minimum value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

minimum and maximum values
In quadratic functions, finding the minimum and maximum values is essential for understanding the graph's behavior. These values indicate the peak or lowest point of a parabola on a graph.
  • If a quadratic function has a minimum value, it means the parabola opens upwards, reaching its lowest point at the vertex.
  • Conversely, if it has a maximum value, the parabola opens downwards and the vertex serves as the highest point.
Values are linked directly to the parabola's shape and direction. The key is identifying whether the parabola opens upwards or downwards based on the quadratic coefficient, often found in the standard quadratic equation form \( ax^2 + bx + c \).
quadratic coefficient
The quadratic coefficient, denoted by \( a \) in the equation \( ax^2 + bx + c \), plays a crucial role in determining the graph's characteristics. It affects both the opening direction and the values that the graph assumes.
  • The sign of \( a \) is vital. A positive \( a \) means the parabola opens upwards, while a negative \( a \) indicates it opens downwards.
  • This coefficient also impacts the steepness or flatness of the parabola. A larger absolute value of \( a \) makes the parabola steeper, while a smaller absolute value results in a wider graph.
In the exercise, the coefficient is \(-\frac{1}{2}\), which directly indicates that the parabola will open downwards.
graph of a parabola
The graph of a parabola is a U-shaped curve that can open either upwards or downwards. This opening direction is determined by the quadratic coefficient. They typically have a vertex, which is the highest or lowest point, depending on how the parabola opens.
  • Upward opening parabolas have a minimum at the vertex.
  • Downward opening parabolas have a maximum at the vertex.
In our original exercise, the graph opens downwards because of the negative quadratic coefficient, forming a U shape with its opening facing the bottom. This means it will have a maximum at the vertex.
opening direction of a parabola
The direction in which a parabola opens is a direct consequence of the quadratic coefficient \( a \). Understanding this is crucial for predicting the parabola's extremum, whether it's a minimum or a maximum value.
  • If \( a > 0 \), the parabola opens upwards, forming a U shape with a minimum point.
  • If \( a < 0 \), the parabola opens downwards, forming an inverted U with a maximum point.
In the given exercise, the coefficient \( a \) is \(-\frac{1}{2}\), indicating that the parabola opens downwards. As such, the graph will feature a maximum value at the vertex.

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Most popular questions from this chapter

Neglecting air resistance, the distance \(s(t)\) in feet traveled by a freely falling object is given by the function \(s(t)=16 t^{2},\) where \(t\) is time in seconds. Use this formula to solve Exercises 79 through \(82 .\) Round answers to two decimal places. The Burj Khalifa, the tallest building in the world, was completed in 2010 in Dubai. It is estimated to be 2717 feet tall. How long would it take an object to fall to the ground from the top of the building? (Source: Council on Tall Buildings and Urban Habitat)

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find the \(y\) -intercept, approximate the \(x\) -intercepts to one decimal place, and sketch the graph. $$ f(x)=2 x^{2}+4 x-1 $$

Evaluate \(\sqrt{b^{2}-4 a c}\) for each set of values. See Section 7.3. $$ a=1, b=-3, c=-1 $$

Find the maximum or minimum value of each function. Approxi- mate to two decimal places. Methane is a gas produced by landfills, natural gas systems, and coal mining that contributes to the greenhouse effect and global warming. Projected methane emissions in the United States can be modeled by the quadratic function $$ f(x)=-0.072 x^{2}+1.93 x+173.9 $$ where \(f(x)\) is the amount of methane produced in million metric tons and \(x\) is the number of years after 2000 . (Source: Based on data from the U.S. Environmental Protection Agency, \(2000-2020\) ) A. According to this model, what will U.S. emissions of methane be in \(2018 ?\) (Round to 2 decimal places.) B. Will this function have a maximum or a minimum? How can you tell? C. In what year will methane emissions in the United States be at their maximum/minimum? Round to the nearest whole year.

Solve by completing the square. See Section 8.1. $$ y^{2}+6 y=-5 $$
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