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Chapter 2: Problem 47

In Exercises \(45-54,\) find the vertex of the parabola associated with each quadratic function. $$f(x)=\frac{1}{3} x^{2}-7 x+5$$

Short Answer

Expert verified
The vertex of the parabola is \((10.5, -31.75)\).

Step by step solution

01

Identify coefficients

In the quadratic function \( f(x) = \frac{1}{3}x^2 - 7x + 5 \), identify the coefficients according to the standard form \( ax^2 + bx + c \). Here, \( a = \frac{1}{3} \), \( b = -7 \), and \( c = 5 \).
02

Use the vertex formula for x-coordinate

The x-coordinate of the vertex of a parabola \( ax^2 + bx + c \) is given by the formula \( x = \frac{-b}{2a} \). Substitute the values of \( a \) and \( b \) to calculate: \( x = \frac{-(-7)}{2 \cdot \frac{1}{3}} = \frac{7}{\frac{2}{3}} = 7 \times \frac{3}{2} = 10.5 \).
03

Calculate the y-coordinate of the vertex

Substitute \( x = 10.5 \) back into the original function to find the \( y \)-coordinate. \( f(10.5) = \frac{1}{3}(10.5)^2 - 7(10.5) + 5 \). Calculate each term: \( \frac{1}{3}(10.5)^2 = \frac{1}{3}(110.25) = 36.75 \), \( -7 \times 10.5 = -73.5 \), and \( +5 \). Calculate \( 36.75 - 73.5 + 5 = -31.75 \).
04

Write the vertex

Combine the results to write the vertex as a coordinate pair. The vertex is \((10.5, -31.75)\).

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Key Concepts

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

Quadratic Function
A quadratic function is a special type of polynomial function where the highest degree of the variable is two. It can usually be expressed in the standard form: \( ax^2 + bx + c \). Here, the letters \( a \), \( b \), and \( c \) are constants, while \( x \) is the variable. When plotted on a graph, this function forms a curve called a parabola.

The coefficient \( a \) determines the direction of the parabola. If \( a \) is positive, the parabola opens upwards, while if \( a \) is negative, it opens downwards.
  • The quadratic function is considered a second-degree polynomial, as the highest power is 2.
  • It consistently produces a U-shaped graph, reflecting its importance in many real-world scenarios.
Knowing the structure of a quadratic function helps you find important features of its graph, like the vertex and axis of symmetry.
Vertex Formula
The vertex of a parabola in a quadratic function \( ax^2 + bx + c \) can be found using the vertex formula. The vertex is a crucial point as it indicates the peak or the lowest point of the parabola, depending on its direction.

To find the \( x \)-coordinate of the vertex, you can use the formula \( x = \frac{-b}{2a} \). Plug in the values of \( a \) and \( b \) from the quadratic equation to compute this coordinate. This formula is derived from completing the square in the quadratic equation.
  • Using the vertex formula efficiently pinpoints the highest or lowest point of the parabola.
  • It simplifies the process by focusing only on the coefficients \( a \) and \( b \).
Once you have the \( x \)-coordinate, substitute it back into the original function to find the \( y \)-coordinate. Together, they describe the vertex as \((x, y)\).
Parabola
Parabolas are the distinctive U-shaped curves formed on a graph by quadratic functions. They come with several fascinating properties that make them a key object of study in mathematics.

Every parabola has a vertex, which is either a maximum or a minimum point, and an axis of symmetry, which is a vertical line that runs through the vertex. This axis acts like a mirror, splitting the parabola into two identical halves.
  • The direction of a parabola depends on the sign of the coefficient \( a \) in the quadratic equation.
  • Parabolas have applications in various fields like physics, engineering, and economics.
The unique properties of parabolas, derived from simple quadratic equations, allow them to model various natural and human-made phenomena effectively. Their symmetrical structure makes them easy to analyze and predict.

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Most popular questions from this chapter

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