/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Find the vertex of the graph of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 2

Find the vertex of the graph of each quadratic function by completing the square or using the vertex formula. $$ f(x)=x^{2}+6 x+5 $$

Short Answer

Expert verified
The vertex is \((-3, -4)\).

Step by step solution

01

Identify the Quadratic Coefficients

Recognize the standard form of the quadratic function: \[ f(x) = ax^2 + bx + c \] Here, \( a = 1 \), \( b = 6 \), and \( c = 5 \).
02

Apply the Vertex Formula

The vertex of a parabola given by the function \( f(x) = ax^2 + bx + c \) can be found using the vertex formula \[ x = -\frac{b}{2a} \]. Substitute \( a = 1 \) and \( b = 6 \) to find \( x \): \[ x = -\frac{6}{2(1)} = -3 \].
03

Substitute to Find the y-Coordinate

Use the x-coordinate found to find the corresponding y-coordinate by substituting back into the function:\[ f(-3) = (-3)^2 + 6(-3) + 5 \].Calculate the result:\[ f(-3) = 9 - 18 + 5 = -4 \].
04

State the Vertex Coordinates

The vertex coordinates are given by the \( x \) value and the corresponding \( y \) value found in previous steps. So, the vertex is \((-3, -4)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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 function that represents a parabola when graphed on a coordinate plane. It is typically expressed in the standard form:
  • \( f(x) = ax^2 + bx + c \)
In this representation:
  • \( a \) is the coefficient of \( x^2 \)
  • \( b \) is the coefficient of \( x \)
  • \( c \) is the constant term
Each of these coefficients plays a pivotal role in determining the shape and position of the parabola:
- If \( a \) is positive, the parabola opens upwards, resembling a "U" shape.- If \( a \) is negative, the parabola opens downwards, forming an inverted "U" shape.
In the given problem, the quadratic function is \( f(x) = x^2 + 6x + 5 \), where \( a = 1 \), \( b = 6 \), and \( c = 5 \). Identifying these coefficients is crucial for solving for the vertex.
Completing the Square
Completing the square is a method used to rewrite a quadratic equation in a "perfect square" form. This helps in identifying the vertex and understanding the function's properties better. To complete the square, consider the quadratic expression \( x^2 + 6x + 5 \).
Follow these simple steps:
  • Take the coefficient of \( x \), which is 6, and divide it by 2 to get 3.
  • Square the result: \( 3^2 = 9 \).
Now, rewrite the expression:
  • Add and subtract 9, forming a perfect square trinomial: \( (x^2 + 6x + 9) - 9 + 5 \).
  • Simplify as: \( (x + 3)^2 - 4 \).
By completing the square, the vertex form \( (x + 3)^2 - 4 \) is easily recognizable, showing a vertex at \((-3, -4)\).
Vertex Formula
The vertex formula provides a straightforward method to find a parabola's vertex without moving to the completing-the-square method. It is rooted in the standard form \( f(x) = ax^2 + bx + c \).
The formula to find the x-coordinate of the vertex is:
  • \( x = -\frac{b}{2a} \)
This formula is derived from calculus and comes from the symmetry of the parabola.
To solve our problem:
  • With \( a = 1 \) and \( b = 6 \), substitute these into the vertex formula: \( x = -\frac{6}{2(1)} = -3 \).
  • Find the y-coordinate by plugging \( x = -3 \) back into the original function: \( f(-3) = (-3)^2 + 6(-3) + 5 = -4 \).
Thus, using the vertex formula, we efficiently find the vertex is \((-3, -4)\).
Parabola
A parabola is a U-shaped curve that is the graph of a quadratic function. It is a symmetrical shape, either opening upwards or downward depending on the sign of the quadratic coefficient \( a \).
  • Symmetry is a key feature, with the axis of symmetry running through the vertex.
  • The vertex is the highest or lowest point depending on the parabola's orientation.
Parabolas in real life can model various phenomena, like the path of a thrown ball or satellite dishes. The parabola of function \( f(x) = x^2 + 6x + 5 \) demonstrated here:
  • Opens upwards, as the value of \( a \) is positive.
  • Has a vertex at \((-3, -4)\), the lowest point on the curve.
Understanding the properties of parabolas helps in predicting the function's behavior and analyzing real-world scenarios.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the quadratic formula to solve each equation. These equations have real number solutions only. $$ m^{2}+5 m-6=0 $$

Use the discriminant to determine the number and types of solutions of each equation. $$ 8 x=3-9 x^{2} $$

For each quadratic equation, choose the correct substitution for \(a, b,\) and \(c\) in the standard form \(a x^{2}+b x+c=0 .\) \(x^{2}+5=-x\) a. \(a=1, b=5, c=-1\) b. \(a=1, b=-1, c=5\) c. \(a=1, b=5, c=1\) d. \(a=1, b=1, c=5\)

Use the quadratic formula to solve each equation. These equations have real number solutions only. $$ \frac{1}{6} x^{2}+x+\frac{1}{3}=0 $$

Solve each inequality. Write the solution set in interval notation. $$ (4 x-9)(2 x+5)<0 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.