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Chapter 5: Problem 20

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. $$ f(x)=-\frac{1}{3} x^{2}-2 x+3 $$

Short Answer

Expert verified
The function has a maximum value of -6 with the axis of symmetry at x = 3.

Step by step solution

01

Identify the Quadratic Form

The given function \( f(x) = -\frac{1}{3}x^2 - 2x + 3 \) is in the standard quadratic form \( ax^2 + bx + c \), where \( a = -\frac{1}{3} \), \( b = -2 \), and \( c = 3 \).
02

Determine if the Function has a Minimum or Maximum Value

Since \( a = -\frac{1}{3} \) and \( a < 0 \), the parabola opens downwards. This means the function has a maximum value.
03

Find the Axis of Symmetry

The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). Substitute \( b = -2 \) and \( a = -\frac{1}{3} \) into the formula: \( x = -\frac{-2}{2(-\frac{1}{3})} = 3 \). So, the axis of symmetry is \( x = 3 \).
04

Find the Maximum Value

To find the maximum value, substitute the axis of symmetry \( x = 3 \) back into the function: \( f(3) = -\frac{1}{3}(3)^2 - 2(3) + 3 \). Simplify to get \( f(3) = -\frac{1}{3}(9) - 6 + 3 = -3 - 6 + 3 = -6 \). Thus, the maximum value is \( -6 \).

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Key Concepts

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

Axis of Symmetry
The axis of symmetry in a quadratic function is a vertical line that divides the parabola into two mirror images, helping us understand the graph's balance. For a quadratic in the form of \( ax^2 + bx + c \), it can be calculated using the formula \( x = -\frac{b}{2a} \). This might seem complex, but in plain words, it's simply about finding the x-coordinate where the parabola bends.
It's what helps us locate the parabola's highest or lowest point on a graph.
  • Use the coefficients \( b \) and \( a \) from the quadratic equation.
  • Substitute these values into the axis of symmetry formula.
  • In our example, \( a = -\frac{1}{3} \) and \( b = -2 \), resulting in \( x = 3 \).
Understanding this concept is key because it gives us vital information about the function's symmetry and helps guide us in both graphing and analyzing its properties.
Maximum Value
A quadratic function can either have a maximum or minimum value, determined by the parabola's orientation. If the parabola opens downwards like a frown, it has a highest point or maximum.
The orientation is known by looking at \(a\):
  • If \( a > 0 \), the curve opens upwards, with a minimum value.
  • If \( a < 0 \), the curve opens downwards, with a maximum value.
In this exercise, \( a = -\frac{1}{3} \) is negative, signifying the parabola has a maximum value.
To find this value, substitute the x-coordinate of the axis of symmetry back into the original function equation. In our solution, at \( x = 3 \), the maximum value occurs, which was calculated to be \( -6 \). This tells us the highest point on the graph is -6 vertically.
Minimum Value
In contrast, when a parabola opens upwards like a smile, it owns a lowest point termed as the minimum value. The curve will inch downwards to this lowest dip efficiently and start escalating again symmetrically.
  • It highlights the least possible value that the quadratic function can achieve.
  • If \( a > 0 \), the parabola smiles with a minimum point present.
  • The minimum value in such a case is positive or zero depending on the position of the curve.
Since our example fallen under a downward opening parabola due to \( a = -\frac{1}{3} \), it conclusively does not exhibit a minimum value. Instead, it shines with a maximum, hence flipping the script of usual minimum value exploration in upward parabolas.
Parabola Orientation
Understanding parabola orientation is like deciphering the emotions of a curve 鈥� whether it celebrates by opening upwards or solemnifies by lowering downwards. This orientation primarily is decided by the sign of \( a \) in the quadratic equation \( ax^2 + bx + c \).
  • A positive \( a \) indicates that the parabola is upward oriented, having a 'smiling' shape.
  • A negative \( a \) signals a downward opening, portraying a 'frowning' parabola.
Our function \( f(x) = -\frac{1}{3}x^2 - 2x + 3 \) employs \( a = -\frac{1}{3} \), a rather negative sign, bravely representing a downward or frowning parabola.
This is a pivotal aspect as it directs us on whether to seek minimum or maximum values and how the graph behaves compared to basic quadratic expectations.

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Most popular questions from this chapter

For the following exercises, use the given information to answer the questions. The weight of an object above the surface of the Earth varies inversely with the square of the distance from the center of the Earth. If a body weighs 50 pounds when it is 3960 miles from Earth's center, what would it weigh it were 3970 miles from Earth's center?

For the following exercises, use the given information to answer the questions. The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 footcandles at a distance of 3 meters. Find the intensity level at 8 meters.

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: \(-4,-1,1,4\) and \((-2, f(-2))=(-2,10)\)

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies inversely as the square of \(x\) and when \(x=1, y=4\).

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the square of \(x .\) When \(x=4\) then \(y=3\). Find \(y\) when \(x=2\).
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