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Chapter 11: Problem 66

Without calculating, tell whether each graph has a minimum value or a maximum value. See the Concept Check in the section. g(x)=-7 x^{2}+x+1

Short Answer

Expert verified
The graph has a maximum value.

Step by step solution

01

Identify the Type of Quadratic Function

The given function is a quadratic function of the form \( g(x) = ax^2 + bx + c \), where \( a = -7 \), \( b = 1 \), and \( c = 1 \). Quadratic functions form parabolas when graphed.
02

Determine the Orientation of the Parabola

The orientation of the parabola is determined by the coefficient \( a \). If \( a > 0 \), the parabola opens upwards, indicating a minimum value. If \( a < 0 \), the parabola opens downwards, indicating a maximum value.
03

Analyze the Coefficient Sign

In the given function, \( a = -7 \), which is less than 0. This means the parabola opens downwards.
04

Conclude the Type of Extrema

Since the parabola opens downwards, the graph of the function has a maximum value.

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Key Concepts

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

Parabola Orientation
In any quadratic function represented as \( f(x) = ax^2 + bx + c \), the term \( ax^2 \) plays the crucial role in determining the orientation of the parabola when graphed. The sign of the coefficient \( a \) dictates how the parabola opens:
  • If \( a > 0 \), the parabola opens upwards. This visual representation resembles a cup facing up.
  • If \( a < 0 \), the parabola opens downwards, much like an upside-down cup.
Understanding the orientation is essential before discussing the function's maximum or minimum values.
In our example function, \( g(x) = -7x^2 + x + 1 \), the coefficient \( a = -7 \) is negative. Therefore, the parabola opens downwards. This initial observation immediately informs us about the presence of a maximum value without needing any calculations.
Maximum and Minimum Values
The orientation of a parabola directly influences whether the quadratic function possesses a maximum or minimum value. Here's how you can predict these values:
  • An upward-opening parabola means there is a minimum value at its vertex since the ends go infinitely upwards.
  • A downward-opening parabola indicates a maximum value at its vertex because the ends infinitely descend.
If you visualize a valley for minimum values and a hill for maximum ones, it's easier to grasp the concept.
For the function \( g(x) = -7x^2 + x + 1 \), we've already noted it opens downwards due to \( a = -7 \). Thus, it has a maximum value at its highest point, at the vertex, without doing any further computation.
Coefficient Analysis
Examining the coefficients in a quadratic equation is key to understanding the behavior of the function. Each coefficient holds specific information:
  • \( a \): Determines the parabola's orientation and whether it has a maximum or minimum.
  • \( b \): Influences the axis of symmetry and the precise location of the vertex, but not the broad orientation itself.
  • \( c \): Represents the y-intercept, indicating where the graph crosses the y-axis.
Analyzing these coefficients allows predictions about the graph's nature even before sketching it.
For the function \( g(x) = -7x^2 + x + 1 \), the coefficient \( a = -7 \), tells us the parabola opens downwards. This alone leads us to conclude there's a maximum value. The other coefficients will find their use during precise plotting or finding the vertex, but they remain secondary to \( a \) for determining the curve's general direction.

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Most popular questions from this chapter

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples 6 and 7 . $$ H(x)=2 x^{2} $$

Sketch the graph of each function. See Section 11.5 $$ g(x)=x+2 $$

Solve by completing the square. See Section 11.1. $$ x^{2}+4 x=12 $$

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}

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ F(x)=-x^{2}+4 $$
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