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Chapter 3: Problem 11

Write each quadratic function in the form \(y=a(x-h)^{2}+k\) and sketch its graph. $$y=x^{2}-3 x$$

Short Answer

Expert verified
The vertex form is \[y = (x - \frac{3}{2})^2 - \frac{9}{4}\].

Step by step solution

01

Identify the quadratic function form

The given quadratic function is in the form of \[y = x^2 - 3x\]The goal is to write it in the vertex form \[y = a(x-h)^2 + k\]
02

Complete the square

To rewrite the function in vertex form, complete the square. Start with \[y = x^2 - 3x\]1. Take half of the coefficient of the linear term \((x\text{-coefficient}) = -3\), which is \[-\frac{3}{2}\]2. Square it: \[-\frac{3}{2}^2 = \frac{9}{4}\]3. Add and subtract this square inside the equation:\[y = x^2 - 3x + \frac{9}{4} - \frac{9}{4}\]
03

Re-write as a perfect square trinomial

Group the perfect square trinomial and the constant separately:\[y = (x^2 - 3x + \frac{9}{4}) - \frac{9}{4}\]Rewrite the trinomial as a square:\[y = (x - \frac{3}{2})^2 - \frac{9}{4}\]
04

Identify the parameters

Compare the equation \[y = (x - \frac{3}{2})^2 - \frac{9}{4}\] with the vertex form \[y = a(x - h)^2 + k\]. Here, \[a = 1\], \[h = \frac{3}{2}\], \[k = -\frac{9}{4}\].

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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 its vertex form. This method is particularly useful because it makes it easier to understand and graph the quadratic function.
To complete the square, follow these steps:
  • Start with the quadratic function in standard form: \[y = ax^2 + bx + c\]
  • Focus on the quadratic and linear terms: \[ax^2 + bx\]
  • Divide all terms by 'a' if 'a' is not equal to 1. For simplicity鈥檚 sake, many basic exercises assume 'a' is 1.
  • Take half of the coefficient of the linear term (the x-coefficient) and then square it. This value will be added and subtracted inside the equation to create a perfect square trinomial.
  • Rewrite the trinomial as a squared binomial.
Let鈥檚 apply this to the given quadratic function, \[y = x^2 - 3x\]. Here, the x-coefficient is -3.
  • Take half of -3: \[ -\frac{3}{2}\]
  • Square it: \[ \left(- \frac{3}{2} \right)^2 = \frac{9}{4}\]
  • Add and subtract this value: \[ y = x^2 - 3x + \frac{9}{4} - \frac{9}{4} \]
  • Group the perfect square trinomial and the constant separately: \[ y = \left( x - \frac{3}{2} \right)^2 - \frac{9}{4} \]
The equation is now in vertex form.
Quadratic Functions
A quadratic function is a second-degree polynomial function of the form \[ y = ax^2 + bx + c \]. The graph of a quadratic function is a parabola that opens upwards if the coefficient 'a' is positive and downwards if 'a' is negative.
The vertex form of a quadratic function is \[ y = a(x-h)^2 + k \], where (h, k) represents the vertex of the parabola.
Understanding the vertex form is beneficial because:
  • It clearly shows the vertex point, making it easier to graph the function.
  • It highlights how the parabola is shifted relative to the origin.
For example, in the equation \[ y = (x - \frac{3}{2})^2 - \frac{9}{4} \], the vertex is at \[ \left( \frac{3}{2}, -\frac{9}{4} \right) \]. The coefficient 'a' (which is 1 in this case) determines the direction (upward or downward) and the width of the parabola.
Rewriting Equations
Rewriting a quadratic function into its vertex form involves completing the square. This process not only helps in expressing the function in a more insightful form but also simplifies the graphing process.
Here's a concise breakdown of the rewriting process applied to the original equation \[ y = x^2 - 3x\]:
  • Identify the terms to complete the square: \[ x^2 - 3x \]
  • Take half of the x-coefficient and square it: \[ \left( -\frac{3}{2} \right)^2 = \frac{9}{4} \]
  • Add and subtract this value: \[ y = x^2 - 3x + \frac{9}{4} - \frac{9}{4} \]
  • Rewrite as a perfect square trinomial: \[ y = \left( x - \frac{3}{2} \right)^2 - \frac{9}{4} \]
By rewriting, you gain a clearer view of the parabolic structure and can swiftly identify key features like the vertex.

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Most popular questions from this chapter

Is the function \(f(x)=x^{2}-4\) one-to-one?

A furniture maker buys foam rubber \(x\) times per year. The delivery charge is 400 dollar per purchase regardless of the amount purchased. The annual cost of storage is figured as 10,000 dollar because the more frequent the purchase, the less it costs for storage. So the annual cost of delivery and storage is given by $$C=400 x+\frac{10,000}{x}$$ a. Graph the function with a graphing calculator. b. Find the number of purchases per year that minimizes the annual cost of delivery and storage.

Solve \(8-6(x-4) \leq 2(x-5)-4(2 x-1)\)

Maximizing Revenue Mona Kalini gives a walking tour of Honolulu to one person for \(\$ 49 .\) To increase her business, she advertised at the National Orthodontist Convention that she would lower the price by \(\$ 1\) per person for each additional person, up to 49 people. a. Write the price per person \(p\) as a function of the number of people \(n\) Minul See Exercise 101 of Section 1.4 b. Write her revenue as a function of the number of people on the tour. Whin Her revenue is the product of \(n\) and \(p\) c. What is the maximum revenue for her tour?

Is the function \(\\{(0,3),(-9,0),(-3,5),(9,7)\\}\) one-to-one?
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