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Chapter 11: Problem 50

Without calculating, tell whether each graph has a minimum value or a maximum value. See the Concept Check in the section. $$ G(x)=3-\frac{1}{2} x+0.8 x^{2} $$

Short Answer

Expert verified
The graph has a minimum value.

Step by step solution

01

Analyze the function type

The function given is a quadratic function as it is of the form \( G(x) = ax^2 + bx + c \). In this case, \( a = 0.8 \), \( b = -\frac{1}{2} \), and \( c = 3 \). Quadratic functions represent parabolas.
02

Determine the direction of the parabola

In quadratic functions, the parabola opens upwards if the coefficient \( a \) is positive, and downwards if \( a \) is negative. Since \( a = 0.8 \) and it is positive, the parabola opens upwards.
03

Identify the presence of a minimum or maximum value

For an upwards-opening parabola, the lowest point of the parabola is called the minimum value. Therefore, this particular quadratic function has a minimum value.

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Key Concepts

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

Understanding Parabolas
When discussing quadratic functions, one of the most significant aspects to understand is the shape of the graph, known as a parabola. A parabola is a symmetrical, curved shape that can open either upwards or downwards. The direction of the opening depends on the quadratic function's leading coefficient: the coefficient of the squared term (i.e., the term containing \(x^2\)). In any quadratic function of the form \( ax^2 + bx + c \), this is the \(a\) value. For instance, in our example, \(a=0.8\). Because this value is positive, we know the parabola opens upwards.
  • If \(a\) is positive, the parabola opens upwards.
  • If \(a\) is negative, the parabola opens downwards.
Understanding the direction of the parabola is crucial in determining whether a graph has a minimum or maximum value.
Determining the Minimum Value
A minimum value of a parabola occurs at its lowest point. This concept is particularly important in fields like physics and economics where optimization is key. When a quadratic function opens upwards, as seen with our example where \(a=0.8\), it indicates the parabola has a minimum value. This is because the lowest point is reached at the vertex.
The vertex of an upwards-opening parabola can be thought of as the 'bottom' of the curve. Mathematically, the vertex is at the point \( x = -\frac{b}{2a} \), and the calculation of the function at this point will give you the minimum value.

  • Minimum value indicates the least point on an upwards-opening parabola.
  • You can find this value using the vertex formula to easily determine the vertex position.
Importance of Coefficients
Coefficients play a vital role in understanding the behavior of quadratic functions. These numbers, which multiply the variables, essentially shape the parabola and determine several of its features. In the equation \( ax^2 + bx + c \), the coefficients \(a\), \(b\), and \(c\) can individually affect the graph:
  • \(a\): Determines the direction and the 'width' or 'narrowness' of the parabola. A larger absolute value of \(a\) results in a narrower parabola.
  • \(b\): Affects the position of the vertex along the horizontal axis. It influences the symmetrically balanced aspect of the graph.
  • \(c\): Represents the y-intercept, the point where the parabola crosses the y-axis. This is the value of the quadratic function when \(x=0\).
Therefore, knowing the coefficients isn't just about filling in numbers. It's about predicting and understanding the shape and position of the parabola on a graph.

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Most popular questions from this chapter

The projected number of Wi-Fi-enabled cell phones in the United States can be modeled by the quadratic function \(c(x)=-0.4 x^{2}+21 x+35,\) where \(c(x)\) is the projected number of Wi-Fi-enabled cell phones in millions and \(x\) is the number of years after 2009 . Round to the nearest million. (Source: Techcrunchies.com) a. Find the number of Wi-Fi-enabled cell phones in the United States for 2010 . b. Find the projected number of Wi-Fi-enabled cell phones in the United States in 2012 . c. If the trend described by this model continues, find the year in which the projected number of \(\mathrm{Wi}\) -Fienabled cell phones in the United States reaches 150 million.

Solve each inequality. Write the solution set in interval notation. $$ \frac{4 x}{x-3} \geq 5 $$

The average total daily supply \(y\) of motor gasoline (in thousands of barrels per day) in the United States for the period \(2000-2008\) can be approximated by the equation \(y=-10 x^{2}+193 x+8464,\) where \(x\) is the number of years after 2000. (Source: Based on data from the Energy Information Administration) a. Find the average total daily supply of motor gasoline in 2004 . b. According to this model, in what year, from 2000 to 2008 , was the average total daily supply of gasoline 9325 thousand barrels per day? c. According to this model, in what year, from 2009 on, will the average total supply of gasoline be 9325 thousand barrels per day?

Use the quadratic formula and a calculator to approximate each solution to the nearest tenth. $$ 3.6 x^{2}+1.8 x-4.3=0 $$

Use the quadratic formula to solve each equation. These equations have real number solutions only. $$ m^{2}+5 m-6=0 $$
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