/*! 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 41 Identify the vertex, axis of sym... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 41

Identify the vertex, axis of symmetry, y-intercept, x-intercepts, and opening of each parabola, then sketch the graph. $$y=x^{2}-3$$

Short Answer

Expert verified
Vertex: (0, -3), Axis of Symmetry: x = 0, y-intercept: (0, -3), x-intercepts: (鈭�3, 0) and (-鈭�3, 0), Opening: upward.

Step by step solution

01

- Identify the Vertex

For a quadratic equation in the form of \( y = ax^2 + bx + c \),the vertex \((h, k)\) can be found using the formula: \( h = -\frac{b}{2a} \). Here, \(a = 1\), \(b = 0\), and \(c = -3\). Substituting in,\( h = -\frac{0}{2*1} = 0 \). Substituting \(x = 0\) into the equation, \( y = 0^2 - 3 = -3 \). Therefore, the vertex is \( (0, -3) \).
02

- Find Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. It can be determined using \( x = h \). From the previous step, we found \( h = 0 \). Therefore, the axis of symmetry is \( x = 0 \).
03

- Determine the y-intercept

The y-intercept is the value of \( y \) when \( x = 0 \). Substituting \( x = 0 \) into the equation, \( y = 0^2 - 3 = -3 \). Therefore, the y-intercept is \( (0, -3) \).
04

- Find x-intercepts

The x-intercepts are the values of \( x \) for which \( y = 0 \). Setting \( y = 0 \) in the equation, \( 0 = x^2 - 3 \), solving for \( x \), we get \( x^2 = 3 \) or \( x = \pm\sqrt{3} \). Therefore, the x-intercepts are \( (\sqrt{3}, 0) \) and \((-\sqrt{3}, 0)\).
05

- Determine the Opening Direction

Since the coefficient of \( x^2 \) is positive \( (a = 1) \), the parabola opens upwards.
06

- Sketch the Graph

To sketch the graph, plot the vertex \( (0, -3) \), axis of symmetry \( x = 0 \), y-intercept \( (0, -3) \), and x-intercepts \((\sqrt{3}, 0) \) and \((-\sqrt{3}, 0) \). Since the parabola opens upwards, draw a smooth curve passing through these points.

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 parabola is the highest or lowest point on its graph, depending on the direction the parabola opens. For a quadratic equation in the form of \( y = ax^2 + bx + c \), the vertex \((h, k)\) can be found using the formula: \( h = -\frac{b}{2a} \). Once you find \( h \), substitute it back into the equation to find \( k \).
In our example, \( y = x^2 - 3 \), we identified that \( a = 1 \), \( b = 0 \), and \( c = -3 \).
The calculations showed that \( h = 0 \), and substituting \( x = 0 \) back into the equation, we found \( y = -3 \).
Thus, the vertex is at \( (0, -3) \).
axis of symmetry
The axis of symmetry is a vertical line that runs through the vertex of the parabola, dividing it into two symmetrical halves. This line can be determined using the equation \( x = h \).
For our quadratic equation \( y = x^2 - 3 \), we found that \( h = 0 \).
Therefore, the axis of symmetry is the line \( x = 0 \), meaning the parabola is symmetric about the y-axis.
y-intercept
The y-intercept is the point where the graph of the parabola crosses the y-axis. This happens when \( x = 0 \).
To find the y-intercept, substitute \( x = 0 \) into the quadratic equation. In our example, substituting \( x = 0 \) into \( y = x^2 - 3 \) gives \( y = -3 \).
Hence, the y-intercept is \( (0, -3) \). This is the same point as the vertex in this particular case.
x-intercepts
The x-intercepts are the points where the graph of the parabola crosses the x-axis. These occur when \( y = 0 \).
To find the x-intercepts, set \( y = 0 \) in the quadratic equation and solve for \( x \). In our equation \( 0 = x^2 - 3 \), this simplifies to \( x^2 = 3 \), yielding \( x = \pm\sqrt{3} \).
Therefore, the x-intercepts are \( (\sqrt{3}, 0) \) and \( (-\sqrt{3}, 0) \). These points show where the parabola meets the x-axis.
parabola direction
The direction in which a parabola opens (upwards or downwards) is determined by the coefficient of \( x^2 \) (denoted as \( a \)) in the quadratic equation.
If \( a > 0 \), the parabola opens upwards.
If \( a < 0 \), the parabola opens downwards.
In our example, \( y = x^2 - 3 \), since \( a = 1 \) (a positive number), the parabola opens upwards. This tells us that the vertex is the lowest point on the graph, and the graph will extend upwards from this point.

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

Find the equation and sketch the graph of each function. A rational function that passes through \((0,5),\) has the line \(y=1\) as a horizontal asymptote, and has the line \(x=2\) as its only vertical asymptote

Find all of the real and imaginary zeros for each polynomial function. $$y=16 x^{3}-33 x^{2}+82 x-5$$

Concert Tickets At \(\$ 10\) per ticket, Willie Williams and the Wranglers will fill all 8000 seats in the Assembly Center. The manager knows that for every \(\$ 1\) increase in the price, 500 tickets will go unsold. a. Write the number of tickets sold \(n\) as a function of ticket price \(p\) Limill See Exercise 102 of Section 1.4 b. Write the total revenue as a function of the ticket price. Giviul Revenue is the product of \(n\) and \(p\) c. What ticket price will maximize the revenue?

Solve each problem. Maximum Volume An open-top box is to be made from a 6 in. by 7 in. piece of copper by cutting equal squares \((x\) in. by \(x\) in.) from each comer and folding up the sides. Write the volume of the box as a function of \(x\). Use a graphing calculator to find the maximum possible volume to the nearest hundredth of a cubic inch.

Find the equation and sketch the graph for each function. A cubic function (a third-degree polynomial function) with \(x\) -intercepts \((2,0),(-3,0),\) and \((4,0)\) and \(y\) -intercept \((0,6)\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.