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,...
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Attached problem- find the vertex of the parabola with work and write in (x,y) form.
![**Problem 3: Finding the Vertex of a Parabola**
Given the quadratic function \( f(x) = 4x^2 + 40x + 104 \), determine the vertex of the parabola described by this equation. Make sure to write the answer in the correct \((x, y)\) form.
*Solution:*
To find the vertex of a parabola given by the quadratic equation \( f(x) = ax^2 + bx + c \), we use the vertex formula:
\[ x = -\frac{b}{2a} \]
1. Identify the coefficients \( a \), \( b \), and \( c \) from the equation:
- \( a = 4 \)
- \( b = 40 \)
- \( c = 104 \)
2. Plug the coefficients into the vertex formula to find the x-coordinate of the vertex:
\[ x = -\frac{40}{2 \cdot 4} = -\frac{40}{8} = -5 \]
3. Substitute \( x = -5 \) back into the original quadratic equation to find the y-coordinate of the vertex:
\[ f(-5) = 4(-5)^2 + 40(-5) + 104 \]
\[ f(-5) = 4(25) + 40(-5) + 104 \]
\[ f(-5) = 100 - 200 + 104 \]
\[ f(-5) = 4 \]
4. Therefore, the vertex of the parabola is:
\[ (-5, 4) \]
The vertex in \((x, y)\) form is \( (-5, 4) \).
Make sure to refer back to guidelines on solving quadratic equations and the vertex form of parabolas for further practice.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29666072-4841-4557-93a0-541aeee2aafd%2Fab59f21f-4043-4380-b365-8e83fd957ac6%2Fzgomby8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3: Finding the Vertex of a Parabola**
Given the quadratic function \( f(x) = 4x^2 + 40x + 104 \), determine the vertex of the parabola described by this equation. Make sure to write the answer in the correct \((x, y)\) form.
*Solution:*
To find the vertex of a parabola given by the quadratic equation \( f(x) = ax^2 + bx + c \), we use the vertex formula:
\[ x = -\frac{b}{2a} \]
1. Identify the coefficients \( a \), \( b \), and \( c \) from the equation:
- \( a = 4 \)
- \( b = 40 \)
- \( c = 104 \)
2. Plug the coefficients into the vertex formula to find the x-coordinate of the vertex:
\[ x = -\frac{40}{2 \cdot 4} = -\frac{40}{8} = -5 \]
3. Substitute \( x = -5 \) back into the original quadratic equation to find the y-coordinate of the vertex:
\[ f(-5) = 4(-5)^2 + 40(-5) + 104 \]
\[ f(-5) = 4(25) + 40(-5) + 104 \]
\[ f(-5) = 100 - 200 + 104 \]
\[ f(-5) = 4 \]
4. Therefore, the vertex of the parabola is:
\[ (-5, 4) \]
The vertex in \((x, y)\) form is \( (-5, 4) \).
Make sure to refer back to guidelines on solving quadratic equations and the vertex form of parabolas for further practice.
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