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Chapter 2: Problem 7

Express \(f(x)\) in the form \(a(x-h)^{2}+k\) $$f(x)=2 x^{2}-12 x+22$$

Short Answer

Expert verified
The vertex form is \( f(x) = 2(x-3)^2 + 4 \).

Step by step solution

01

Identify the Quadratic Function

You are given a quadratic function in the form of \(f(x) = 2x^2 - 12x + 22\). We want to express it in the vertex form \(f(x) = a(x - h)^2 + k\).
02

Extract the Coefficient 'a'

In the given equation \(f(x) = 2x^2 - 12x + 22\), the coefficient \(a\) of \(x^2\) is 2. This will remain unchanged in the vertex form.
03

Complete the Square

To complete the square, consider the expression \(2(x^2 - 6x)\). Half the coefficient of \(x\) (which is -6) is -3, and it's square is 9. So, add and subtract 9 inside the bracket:1. Rewrite the expression as \(2(x^2 - 6x + 9 - 9)\).2. Simplify to \(2((x-3)^2 - 9)\).
04

Simplify the Expression

Multiply through by the factor 2: \(2((x-3)^2 - 9) = 2(x-3)^2 - 18\). Thus, the original quadratic becomes \(f(x) = 2(x-3)^2 - 18 + 22\).
05

Simplify and Write in Vertex Form

Combine the constants \(-18 + 22\) to get 4.Hence, \(f(x) = 2(x-3)^2 + 4\).This is the vertex form of the given function.

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Key Concepts

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

Completing the Square
Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This process helps rewrite quadratic functions in a way that reveals the vertex form, which is particularly useful for graphing.

To complete the square for a quadratic equation like \( ax^2 + bx + c \), follow these steps:
  • Factor out the coefficient of \( x^2 \) if it isn't 1. In our example, it is already 2, so we work with \( 2(x^2 - 6x) \).
  • Take half of the \( b \) coefficient (where \( b = -6 \) in our case), square it, and add/subtract it within the bracket. Half of \(-6\) is \(-3\), and squaring \(-3\) gives you \(9\).
  • Insert \( +9 \) and \( -9 \) inside your equation to keep the function's value unchanged (\( 2(x^2 - 6x + 9 - 9) \)).
  • Rewrite the perfect square trinomial \((x^2 - 6x + 9)\) as \((x - 3)^2\).
After completing these steps, you create a form that brings clarity about the graph鈥檚 shape and position by finding the vertex.
Quadratic Function
A quadratic function is any function that can be described by the standard form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These functions graph as a parabola, which can open upwards or downwards depending on the sign of \( a \).

Key features of quadratic functions include:
  • The vertex, which is either the highest or lowest point on the graph.
  • The axis of symmetry, a vertical line that splits the parabola into two mirror-image halves.
  • The direction the parabola opens, dictated by the sign of \( a \). If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.
Understanding these features allows you to deduce more about the function's real-world applications and graph it effectively.
Vertex Form Transformation
Transforming a quadratic function into vertex form involves rearranging it to \( f(x) = a(x-h)^2 + k \). This form highlights the vertex of the parabola, given by the point \((h, k)\).

The main advantages of the vertex form are:
  • It makes identifying the vertex straightforward, which helps in graphing the parabola efficiently.
  • It provides a clear picture of how the parabola shifts from its basic position \( y = ax^2 \), due to \( h \) (horizontal shift) and \( k \) (vertical shift).
  • It gives insights into maximum or minimum values directly depending on whether the parabola opens upward or downward.
In our exercise, expressing \( f(x) = 2x^2 - 12x + 22 \) as \( f(x) = 2(x-3)^2 + 4 \) neatly gives the vertex \((3, 4)\), offering a clear interpretation of the function's characteristics.

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Most popular questions from this chapter

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