/*! 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 15 Graph the function, label the ve... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 15

Graph the function, label the vertex, and draw the axis of symmetry. $$ g(x)=(x+1)^{2} $$

Short Answer

Expert verified
Vertex: \( (-1, 0) \), Axis of symmetry: \( x = -1 \).

Step by step solution

01

Identify the Vertex

For the function \( g(x) = (x+1)^2 \), the vertex form of a quadratic function is given by \( g(x) = a(x-h)^2 + k \). Here, \( a = 1 \), \( h = -1 \), and \( k = 0 \). Therefore, the vertex of the function is at \( (-1, 0) \).
02

Draw the Axis of Symmetry

The axis of symmetry for a quadratic function in the form \( g(x) = a(x-h)^2 + k \) is the vertical line that passes through the vertex. Thus, the axis of symmetry is \( x = -1 \).
03

Plot the Vertex and Axis of Symmetry

On a coordinate plane, mark the vertex at the point \( (-1, 0) \). Then draw the vertical line \( x = -1 \) to represent the axis of symmetry.
04

Graph the Function

Since \( g(x) = (x+1)^2 \) is a standard parabola that opens upwards (because the coefficient of \( (x+1)^2 \) is positive), plot the parabola by plotting additional points around the vertex, such as \( (0, 1) \) and \( (-2, 1) \). Connect these points with a smooth curve to complete the graph.

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
The vertex of a quadratic function is a key feature because it represents the maximum or minimum point of the parabola. For the function given, \( g(x) = (x+1)^2 \), the vertex form of a quadratic equation is \( g(x) = a(x-h)^2 + k \). Here, \( a = 1 \), \( h = -1 \), and \( k = 0 \). Therefore, the vertex is at the point \( (-1, 0) \). To find the vertex, we need to identify these variables clearly:
  • \( h \) controls the horizontal shift
  • \( k \) controls the vertical shift
  • \( a \) dictates the direction (upward or downward) and the width of the parabola
The vertex can help in graphing the quadratic function as it gives a central point from where the parabola expands.
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It always runs through the vertex of the parabola. For our quadratic function \( g(x) = (x+1)^2 \), the axis of symmetry is given by the equation \( x = h \), where \( h \) is the horizontal shift.
The vertex we found earlier is at \( (-1, 0) \), so the axis of symmetry is the line \( x = -1 \). This line helps in ensuring that our graph reflects the correct shape and symmetry. When graphing, you can draw this vertical line to visually confirm the symmetry of your plotted points and the curve.
Parabola
A parabola is the shape of the graph for any quadratic function. It can either open upwards or downwards based on the value of \( a \):
  • If \( a \) is positive, the parabola opens upwards.
  • If \( a \) is negative, the parabola opens downwards.
For the function \( g(x) = (x+1)^2 \), since \( a = 1 \) (which is positive), the parabola opens upwards. The general shape of a parabola is determined by plotting points symmetrically on either side of the vertex. For example, besides the vertex \( (-1, 0) \), you could plot points like \( (0, 1) \) and \( (-2, 1) \). These points help in defining the curve more accurately.
Finally, connect these points smoothly to complete the graph of the function, ensuring the curve bends symmetrically around the axis of symmetry.

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

Solve. (Use \(\left.4.9 t^{2}+v_{0} t=s .\right)\) a) A life preserver is dropped from a helicopter at an altitude of \(75 \mathrm{m} .\) Approximately how long does it take the life preserver to reach the water? b) A coin is tossed downward with an initial velocity of \(30 \mathrm{m} / \mathrm{sec}\) from an altitude of \(75 \mathrm{m}\) Approximately how long does it take the coin to reach the ground? c) Approximately how far will an object fall in \(2 \sec ,\) if thrown downward at an initial velocity of \(20 \mathrm{m} / \mathrm{sec}\) from a helicopter?

Find each solution set. $$\left|\frac{x+2}{x-1}\right| \leq 3$$

Solve. During the first part of a trip, Tara drove \(120 \mathrm{mi}\) at a certain speed. Tara then drove another \(100 \mathrm{mi}\) at a speed that was \(10 \mathrm{mph}\) slower. If the total time of Tara's trip was 4 hr, what was her speed on each part of the trip?

Trajectory of a Launched Object. The height above the ground of a launched object is a quadratic function of the time that it is in the air. Suppose that a flare is launched from a cliff 64 ft above sea level. If 3 sec after being launched the flare is again level with the cliff, and if 2 sec after that it lands in the sea, what is the maximum height that the flare reaches?

Minimizing Area. A 36-in. piece of string is cut into two pieces. One piece is used to form a circle while the other is used to form a square. How should the string be cut so that the sum of the areas is a minimum?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

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.