1 x² + 3x 3 identify the axis of symmetry and the vertex of the corresponding parabola. Show Bring the quadratic function f(x) 5 into the standard form and your work!
1 x² + 3x 3 identify the axis of symmetry and the vertex of the corresponding parabola. Show Bring the quadratic function f(x) 5 into the standard form and your work!
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![### Quadratic Functions and Parabolas
#### Problem 2
**(a)** Bring the quadratic function \( f(x) = -\frac{1}{3}x^2 + 3x - 5 \) into the standard form and identify the axis of symmetry and the vertex of the corresponding parabola. Show your work!
**Solution:**
1. Begin with the given quadratic function:
\[ f(x) = -\frac{1}{3}x^2 + 3x - 5 \].
2. To convert the quadratic function into its standard form \( f(x) = a(x-h)^2 + k \), we complete the square.
3. Factor out the coefficient of \( x^2 \) from the first two terms:
\[ f(x) = -\frac{1}{3}(x^2 - 9x) - 5 \].
4. Complete the square inside the parentheses. Take half of the coefficient of \( x \), square it, and add and subtract this value inside the parentheses:
\[ x^2 - 9x \]
\[ = x^2 - 9x + \left(\frac{9}{2}\right)^2 - \left(\frac{9}{2}\right)^2 \]
\[ = x^2 - 9x + \frac{81}{4} - \frac{81}{4} \]
\[ = \left(x - \frac{9}{2}\right)^2 - \frac{81}{4} \]
5. Substitute back into the function and simplify:
\[ f(x) = -\frac{1}{3} \left[\left(x - \frac{9}{2}\right)^2 - \frac{81}{4}\right] - 5 \]
\[ = -\frac{1}{3} \left(x - \frac{9}{2}\right)^2 + \frac{81}{12} - 5 \]
\[ = -\frac{1}{3} \left(x - \frac{9}{2}\right)^2 + \frac{81}{12} - \frac{60}{12} \]
\[ = -\frac{1}{3} \left(x - \frac{9}{2}\right)^2 + \frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2ea4237-947f-4fb7-b8d4-fb858663c4d1%2Fe35bac77-7408-48bc-bddc-ca41020edf35%2F6rb537p.jpeg&w=3840&q=75)
Transcribed Image Text:### Quadratic Functions and Parabolas
#### Problem 2
**(a)** Bring the quadratic function \( f(x) = -\frac{1}{3}x^2 + 3x - 5 \) into the standard form and identify the axis of symmetry and the vertex of the corresponding parabola. Show your work!
**Solution:**
1. Begin with the given quadratic function:
\[ f(x) = -\frac{1}{3}x^2 + 3x - 5 \].
2. To convert the quadratic function into its standard form \( f(x) = a(x-h)^2 + k \), we complete the square.
3. Factor out the coefficient of \( x^2 \) from the first two terms:
\[ f(x) = -\frac{1}{3}(x^2 - 9x) - 5 \].
4. Complete the square inside the parentheses. Take half of the coefficient of \( x \), square it, and add and subtract this value inside the parentheses:
\[ x^2 - 9x \]
\[ = x^2 - 9x + \left(\frac{9}{2}\right)^2 - \left(\frac{9}{2}\right)^2 \]
\[ = x^2 - 9x + \frac{81}{4} - \frac{81}{4} \]
\[ = \left(x - \frac{9}{2}\right)^2 - \frac{81}{4} \]
5. Substitute back into the function and simplify:
\[ f(x) = -\frac{1}{3} \left[\left(x - \frac{9}{2}\right)^2 - \frac{81}{4}\right] - 5 \]
\[ = -\frac{1}{3} \left(x - \frac{9}{2}\right)^2 + \frac{81}{12} - 5 \]
\[ = -\frac{1}{3} \left(x - \frac{9}{2}\right)^2 + \frac{81}{12} - \frac{60}{12} \]
\[ = -\frac{1}{3} \left(x - \frac{9}{2}\right)^2 + \frac
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education