/*! 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 4 Find the vertex, the \(x\) -inte... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 4

Find the vertex, the \(x\) -intercepts (if any), and sketch the parabola. \(f(x)=x^{2}+6 x+9\)

Short Answer

Expert verified
The vertex of the parabola is \((-3, 0)\), there is a single x-intercept at \((-3, 0)\), and the parabola opens upwards, symmetric around the line x = -3.

Step by step solution

01

Identify the form of the quadratic function

The given quadratic function is: \(f(x) = x^2 + 6x + 9\) This is in the standard form of a quadratic function, which is: \(f(x) = ax^2 + bx + c\) In our case, \(a = 1\), \(b = 6\), and \(c = 9\).
02

Find the vertex of the parabola

The vertex of a parabola in standard form is given by the formula: \((h, k) = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\) We already know the values for a and b, so we can plug in to find the x-coordinate (h) of the vertex: \(h = -\frac{6}{2(1)} = -\frac{6}{2} = -3\) Now, we'll find the y-coordinate (k) of the vertex by plugging h into the function: \(k = f(-3) = (-3)^{2} + 6(-3) + 9 = 9 - 18 + 9 = 0\) So, the vertex of the parabola is \((-3, 0)\).
03

Find the x-intercepts (if any)

The x-intercepts are the points where the parabola touches the x-axis, which means the y-coordinate (f(x)) will be 0. We can set up the equation: \(0 = x^{2} + 6x + 9\) We can try factoring this equation or using the quadratic formula to find the x-intercepts. In this case, factoring is more straightforward: \(0 = (x + 3)^{2}\) As we have a perfect square, there is only one x-intercept, which is the double root: \(x = -3\) So, the x-intercept is \((-3, 0)\).
04

Sketch the parabola

To sketch the parabola, we'll start by marking the vertex and the x-intercepts. We know the vertex is (-3, 0) and the x-intercept is (-3, 0), which is the same point. Since the leading coefficient 'a' is positive, the parabola will open upwards. Now, we can sketch the parabola with the vertex at \((-3, 0)\), opening upwards, and passing through the x-intercept at \((-3, 0)\). The resulting parabola is symmetric around the vertical line x = -3. So, the vertex is \((-3, 0)\), there is a single x-intercept at \((-3, 0)\), and the parabola opens upwards, symmetric around the line x = -3.

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.

Vertex of a Parabola
The vertex of a parabola is a crucial point, often denoted by \((h, k)\), which represents the highest or lowest point on the graph, depending on the parabola's orientation. This point provides a key insight into the parabola's axis of symmetry and general shape.
You can find the vertex of a parabola in standard form \(f(x) = ax^2 + bx + c\) using the formula:
  • For the x-coordinate (\(h\)): \(h = -\frac{b}{2a}\).
  • Substitute this value back into the function to find the y-coordinate (\(k\)): \(k = f(h)\).
In the example \(f(x) = x^2 + 6x + 9\), we plug in the values for \(b\) and \(a\): \[h = -\frac{6}{2(1)} = -3\]Then, substitute \(h\) into the function: \[k = (-3)^2 + 6(-3) + 9 = 0\]Thus, the vertex is \((-3, 0)\), meaning the parabola has its lowest point at this coordinate and is symmetrical about the line \(x = -3\).
X-Intercepts
X-intercepts are where the graph of the parabola intersects the x-axis, which means the output, or \(f(x)\), equals zero at these points. Solving for x-intercepts gives insight into the roots of the quadratic equation.
To find them, set the quadratic equation \(f(x) = ax^2 + bx + c\) to zero and solve for \(x\):
  • If factorizable, rewrite the equation as products and solve each factor for zero.
  • Alternatively, use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In our case, the quadratic \(x^2 + 6x + 9 = 0\) factors into:\[(x + 3)^2 = 0\]This means there's a "double root" at \(x = -3\). The single x-intercept is \((-3, 0)\), representing both a vertex and a root. This scenario is typical for perfect square quadratics, leading to the graph only touching the x-axis at one point.
Sketching Parabolas
Sketching a parabola is a simple yet powerful visualization technique that brings the algebraic function to life on a graph. Here鈥檚 how you can sketch the parabola using its vertex and intercepts.
Start by marking essential points:
  • Draw the vertex, in this case, \((-3, 0)\), onto your coordinate plane.
  • Identify and plot x-intercepts; here, it overlaps with the vertex at \((-3, 0)\).
Understand the parabola's orientation:
  • If the leading coefficient \(a > 0\), the parabola opens upwards. Here \(a = 1\), so the parabola opens upwards.
  • A parabola opens downward if \(a < 0\).
Consider symmetry:
  • Note that the parabola is symmetrical about the line \(x = -3\), which runs vertically through the vertex.
Finally, draw the curve by sketching smooth, symmetric arms extending from the vertex. These should move away from the x-axis, depending on the parabola's wideness or narrowness, typically influenced by the value of \(a\). This gives you a visual representation of the quadratic relationship, showing its shape and direction based on calculated features.

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

Social Security recipients receive an automatic cost-of-living adjustment (COLA) once each year. Their monthly benefit is increased by the same percentage that consumer prices have increased during the preceding year. Suppose consumer prices have increased by 5.3\% during the preceding year. a. Express the adjusted monthly benefit of a Social Security recipient as a function of his or her current monthly benefit. b. If Carlos Garcia's monthly Social Security benefit is now \(\$ 1020\), what will be his adjusted monthly benefit?

Determine whether the given function is a polynomial function, a rational function, or some other function. State the degree of each polynomial function. \(G(x)=2\left(x^{2}-3\right)^{3}\)

Find the points of intersection of the graphs of the functions. \(f(x)=x^{2}-2 x-2 ; g(x)=-x^{2}-x+1\)

DruG DosaGES A method sometimes used by pediatricians to calculate the dosage of medicine for children is based on the child's surface area. If \(a\) denotes the adult dosage (in milligrams) and if \(S\) is the child's surface area (in square meters), then the child's dosage is given by $$ D(S)=\frac{S a}{1.7} $$ a. Show that \(D\) is a linear function of \(S\). Hint: Think of \(D\) as having the form \(D(S)=m S+b\). What are the slope \(m\) and the \(y\) -intercept \(b\) ? b. If the adult dose of a drug is \(500 \mathrm{mg}\), how much should a child whose surface area is \(0.4 \mathrm{~m}^{2}\) receive?

Find the vertex, the \(x\) -intercepts (if any), and sketch the parabola. \(f(x)=3 x^{2}-5 x-2\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.