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Chapter 11: Problem 8

Find the vertex of the graph of each quadratic function. See Examples I through 4 $$ f(x)=x^{2}-9 x+8 $$

Short Answer

Expert verified
The vertex is \( \left(\frac{9}{2}, -\frac{49}{4}\right) \).

Step by step solution

01

Understand the Vertex Form of a Quadratic

The vertex form of a quadratic function is given by \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. Our task is to convert the given quadratic function \( f(x) = x^2 - 9x + 8 \) into this form to identify the vertex.
02

Complete the Square

To convert \( f(x) = x^2 - 9x + 8 \) into vertex form, complete the square for the quadratic expression. Start by focusing on \( x^2 - 9x \). Take half of the coefficient of \( x \), which is \(-9/2\), and square it to get \((9/2)^2 = 81/4\). Add and subtract this term inside the function for completing the square: \( f(x) = (x^2 - 9x + \frac{81}{4}) - \frac{81}{4} + 8 \).
03

Simplify the Expression

The expression \((x^2 - 9x + \frac{81}{4})\) becomes \((x - \frac{9}{2})^2\), forming a perfect square. Thus, our function becomes \( f(x) = (x - \frac{9}{2})^2 - \frac{81}{4} + 8 \). Simplify the constant terms: \(-\frac{81}{4} + 8 = -\frac{81}{4} + \frac{32}{4} = -\frac{49}{4}\). So, \( f(x) = (x - \frac{9}{2})^2 - \frac{49}{4} \).
04

Identify the Vertex

Now that the function is in vertex form \( f(x) = (x - \frac{9}{2})^2 - \frac{49}{4} \), the vertex \( (h,k) \) is \((\frac{9}{2}, -\frac{49}{4}) \). Hence, the vertex of the quadratic function \( f(x) = x^2 - 9x + 8 \) is \( \left(\frac{9}{2}, -\frac{49}{4}\right) \).

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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 type of polynomial function that is characterized by its highest exponent being 2. It can be generally written as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero.
A very recognizable feature of a quadratic function is its graph, which is a U-shaped curve called a parabola.
Quadratic functions are crucial in many mathematical applications because they model various physical phenomena.
  • Parabolas are widely seen in physics, such as the paths of projectiles.
  • They can also represent economic functions like cost or revenue projections.
  • When analyzing a quadratic function, key properties include the direction of the parabola (which way it opens), the vertex, and the axis of symmetry.
The constant \( a \) in the function affects the opening direction and width of the parabola. If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards.
Vertex Form
The vertex form of a quadratic function makes it easy to identify the vertex of the parabola. This form is \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex.
Transforming a quadratic function into vertex form involves some handy math steps. The vertex form highlights the most critical feature of the parabola, which is its turning point.
  • The turning point or vertex is crucial as it represents the maximum or minimum of the parabola.
  • In vertex form, the number \( h \) shifts the graph horizontally, while \( k \) shifts it vertically.
  • Understanding vertex form also links nicely with transformations in broader algebra.
This form is particularly beneficial when you're looking to maximize or minimize quadratic functions, such as in optimization problems.
Completing the Square
Completing the square is a method used to transform a standard quadratic function into vertex form. It involves turning a part of the quadratic expression into a perfect square trinomial.
This is a strategic process to simplify complex quadratic equations, especially when you need to find the vertex.
  • You start by identifying and factoring the quadratic coefficient.
  • Next, take half of the linear coefficient, square it, and add it inside the square.
  • You must also subtract this same value to maintain equivalence in the original equation.
This process looks daunting, but with practice, it becomes a straightforward trick in the algebra toolkit.
For instance, changing \( x^2 - 9x + 8 \) into vertex form involves adding and subtracting \( (9/2)^2 \) within the equation, enabling us to rewrite it as a squared term plus a constant.
Parabola
A parabola is the graph of a quadratic function. It is a symmetric curve shaped like a bowl. Parabolas have several important features, including a vertex, an axis of symmetry, and direction.
Understanding these features helps in analyzing and graphing quadratic functions effectively.
  • The vertex is the point where the parabola changes direction. It is either the maximum or minimum point depending on the parabola's orientation.
  • The axis of symmetry is a vertical line that passes through the vertex, and it divides the parabola into two symmetrical halves.
  • The direction of the parabola is determined by the sign of the coefficient \( a \): positive \( a \) means it opens upwards; negative means downward.
When illustrating a parabola, noticing these details makes plotting parabolas much easier.
For example, knowing that the vertex of \( f(x) = (x - \frac{9}{2})^2 - \frac{49}{4} \) is \( (\frac{9}{2}, -\frac{49}{4}) \) gives you a key point to start constructing the graph accurately.

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Most popular questions from this chapter

Solve. See the Concept Check in this section. Which description of \(f(x)=-213(x-0.1)^{2}+3.6\) is correct? $$ \begin{array}{|l|l|} \hline \text { Graph Opens } & {\text { Vertex }} \\ \hline \text { a. upward } & {(0.1,3.6)} \\ \hline \text { b. upward } & {(-213,3.6)} \\ \hline \text { c. downward } & {(0.1,3.6)} \\ \hline \text { d. downward } & {(-0.1,3.6)} \\ \hline \end{array} $$

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

Find the maximum or minimum value of each function. Approximate to two decimal places. \(f(x)=2.3 x^{2}-6.1 x+3.2\)

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Example 8 . $$ g(x)=4(x-4)^{2}+2 $$

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. See Section 11.1. $$ x^{2}-10 x $$
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