/*! 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 24 (a) find the vertex and axis of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 24

(a) find the vertex and axis of symmetry of each quadratic function. (b) Determine whether the graph is concave up or concave down. (c) Graph the quadratic function. \(f(x)=-(x+4)^{2}-1\)

Short Answer

Expert verified
Vertex: (-4, -1). Axis of Symmetry: x = -4. Concave down.

Step by step solution

01

Identify the Quadratic Function Form

The given quadratic function is in the vertex form: \(f(x) = a(x-h)^2 + k\). Compare the given function \(f(x) = -(x+4)^2 - 1\) to the vertex form.
02

Determine the Vertex

From \(f(x) = -(x+4)^2 - 1\), we can see that \(a = -1\), \(h = -4\), and \(k = -1\). Therefore, the vertex of the function is \((-4, -1)\).
03

Find the Axis of Symmetry

The axis of symmetry for a quadratic function in vertex form \(a(x-h)^2 + k\) is given by \(x = h\). Thus, the axis of symmetry is \(x = -4\).
04

Determine Concavity

The concavity of the parabola is determined by the coefficient \(a\). If \(a > 0\), the graph is concave up; if \(a < 0\), the graph is concave down. Here, \(a = -1\), so the graph is concave down.
05

Graph the Function

To graph \(f(x) = -(x+4)^2 - 1\), plot the vertex at \((-4, -1)\) and use the information about concavity and the axis of symmetry to draw the parabolic curve that opens downwards.

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 form
Understanding the vertex form of a quadratic function is crucial when analyzing and graphing these types of equations. The vertex form is given by a(x - h)^2 + k where:

    • is the coefficient that determines the width and direction of the parabola.
  • h
    • is the x-coordinate of the vertex.
  • k
    • is the y-coordinate of the vertex.
In this problem, you're given the functionf(x)=-(x+4)^2-1. This matches the vertex form if you recognize that it can be written asa(x - (-4))^2 + (-1). Hence, -4 is h, and -1 is k. The vertex of the function is (-4, -1).The vertex is a critical point on the graph as it represents the highest or lowest point of the parabola depending on its concavity.
axis of symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror images.For any quadratic function in vertex form a(x - h)^2 + k, the axis of symmetry can be found using x = h. In this case, our quadratic function isf(x)=-(x+4)^2-1. From the vertex form, we identifiedh = -4, so the axis of symmetry isx = -4.Graphically, this means if you draw a vertical line atx = -4, the parabola will be symmetric about this line. When graphing, this essentially helps ensure the left and right sides of the parabola match perfectly.
concavity
Concavity tells us the direction in which the parabola opens. This is determined by the coefficient a in the vertex form. If a > 0, the parabola opens upwards, creating a 'U' shape. Conversely, if a < 0, the parabola opens downwards, like an upside-down 'U'.In our problem,a = -1, which is less than 0. Therefore, the parabola is concave down. It's important to note that the vertex will be the highest point when the parabola is concave down. This implies the value at the vertex is the maximum value of the function.This concept will help you anticipate the overall shape of the graph before plotting the points.
graphing parabolas
Graphing a quadratic function involves several key steps, which we have already discussed: identifying the vertex, axis of symmetry, and concavity. Now let's put it all together.
  • Start by plotting the vertex. For f(x)=-(x+4)^2-1, the vertex is (-4, -1).
    • Next, draw the axis of symmetry as a dashed vertical line through x = -4. Now use the fact that the parabola is concave down to sketch a curve opening downwards from the vertex.

To ensure accuracy, you can plot additional points by choosing x-values around the vertex and calculating their corresponding y-values.This step-by-step approach will allow you to graph the quadratic function comprehensively.

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

(a) Graph fand g on the same Cartesian plane. (b) Solve \(f(x)=g(x)\) (c) Use the result of part (b) to label the points of intersection of the graphs of fand \(g\). (d) Shade the region for which \(f(x)>g(x)\); that is, the region below fand above \(g\). \(f(x)=-x^{2}+9 ; \quad g(x)=2 x+1\)

(a) Graph fand \(g\) on the same Cartesian plane. (b) Solve \(f(x)=g(x)\) (c) Use the result of part (b) to label the points of intersection of the graphs of fand \(g\). (d) Shade the region for which \(f(x)>g(x)\); that is, the region below fand above \(g\). \(f(x)=-x^{2}+7 x-6 ; \quad g(x)=x^{2}+x-6\)

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=-5 x^{2}+20 x+3\)

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=x^{2}-2 x-3\)

The distance \(d\) between the bottom of a suspended spring and a countertop is a linear function of the weight \(w\) attached to the bottom of the spring. The bottom of the spring is 9 inches from the countertop whenthe attached weight is 1.5 pounds and 5 inches from the countertop when the attached weight is 2.5 pounds. (a) Find a linear model that relates the distance \(d\) from the countertop and the weight \(w\). (b) Find the distance between the bottom of the spring and the countertop if no weight is attached. (c) What is the smallest weight that will make the bottom of the spring reach the countertop? (Ignore the thickness of the weight.)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

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.