/*! 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 3 Plot the graph of each equation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 3

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x=-4 y^{2}-1\)

Short Answer

Expert verified
Graph is a left-opening parabola with vertex at (-1, 0) and symmetrical about the x-axis.

Step by step solution

01

Understand the Equation

The equation given is \(x = -4y^2 - 1\). This is a form of a quadratic equation in terms of \(y\), similar to \(y = ax^2 + bx + c\), but the role of \(x\) and \(y\) are swapped.
02

Check for Symmetries

Since the equation is of the form \(x = a y^2 + c\), it represents a parabola. Parabolas of this form open left or right. Test for symmetry:- Symmetry with respect to the x-axis: Substitute \(y = -y\), and check if the equation remains equivalent. \(x = -4(-y)^2 - 1 = -4y^2 - 1\) (True). Thus, the graph is symmetric with respect to the x-axis.
03

Find the x-intercepts

To find the x-intercept, set \(y = 0\) in the equation: \[x = -4(0)^2 - 1 = -1\]So, the x-intercept is \((-1, 0)\).
04

Locate the y-intercepts

The equation is quadratic in terms of \(y\), meaning there could be no y-intercepts (unless the parabola intersects the y-axis) because it's solved for \(x\). We need to set \(x = 0\) and solve for \(y\): \[0 = -4y^2 - 1\]The equation becomes \(-4y^2 = 1\), leading to a non-existent real solution as it results in \(y^2 = -\frac{1}{4}\). Therefore, there are no real y-intercepts.
05

Sketch the Graph

The parabola opens to the left (since the coefficient \(-4\) is negative).- The vertex of the parabola can be found by setting \(y = 0\), giving vertex \(V = (-1, 0)\).- Because the parabola is symmetric about the x-axis and the x-intercept is at \((-1,0)\), this is the starting point.Drawing this, you should aim for a symmetrical parabola opening to the left.

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.

Parabola Symmetry
When graphing a quadratic equation, one essential aspect to consider is the symmetry of the parabola. For the equation \( x = -4y^2 - 1 \), the variables \( x \) and \( y \) are swapped compared to the usual quadratic equation. This equation represents a parabola that can be analyzed for symmetry through substitution. By substituting \( y = -y \) into the equation, the form remains unchanged: \( x = -4(-y)^2 - 1 = -4y^2 - 1 \). This indicates that the graph is symmetric with respect to the x-axis.

This x-axis symmetry is typical of parabolas of the form \( x = ay^2 + c \). Such parabolas open either to the left or right, unlike the more familiar upward or downward opening parabolas seen in \( y = ax^2 + bx + c \). Knowing this symmetry helps us predict the shape and direction of the parabola.
  • Symmetry helps simplify graphing.
  • Understanding which axis the graph is symmetrical to is crucial.
X-Intercepts
To find where a parabola crosses the x-axis, we look for the x-intercepts by setting \( y = 0 \). This step provides points on the graph where the curve touches or crosses the x-axis. For our equation \( x = -4y^2 - 1 \), setting \( y = 0 \) simplifies the equation to \( x = -1 \). Thus, the x-intercept is found at the point \((-1, 0)\).

An x-intercept is a valuable point of reference when sketching a graph, as it represents one of the points where the parabola intersects the x-axis. This intercept is significant because it establishes the lowest or highest point along the x-axis that our curve will touch.
  • The x-intercept is obtained by letting \( y = 0 \).
  • It provides a specific point crucial for plotting the graph accurately.
Quadratic Equation Analysis
Analyze the quadratic equation, \( x = -4y^2 - 1 \), to understand its characteristics fully. The parabolic equation shows features that include direction, width, and vertex position. The coefficient \(-4\) in \(-4y^2\) is most telling; it indicates the parabola opens to the left since it's negative. The higher the absolute value of this coefficient, the narrower the parabola.

This equation, formulated differently from the standard form, teaches us about key vertex properties and curvatures. Although the vertex is not explicitly calculated by a formula, knowing the x-intercept at \((-1, 0)\) helps us visualize it as the vertex. The vertex represents the starting point or the peak along the axis of symmetry.
  • Direction becomes clear via the coefficient sign.
  • Width is linked to the size of the coefficient.
  • Vertices are notable points where curve properties hinge.
By examining these features, along with intercepts and symmetry, you can successfully plot a parabola, grasping the entirety of its geometric behavior.

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

Write \(p(x)=1 / \sqrt{x^{2}+1}\) as a composite of four functions.

29\. Starting at noon, airplane A flies due north at 400 miles per hour. Starting 1 hour later, airplane \(\mathrm{B}\) flies due east at 300 miles per hour. Neglecting the curvature of the Earth and assuming that they fly at the same altitude, find a formula for \(D(t),\) the distance between the two airplanes \(t\) hours after noon. Hint: There will be two formulas for \(D(t),\) one if \(0

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(y=\frac{x}{x^{2}+1}\)

We now explore the relationship between \(A \sin (\omega t)+\) \(B \cos (\omega t)\) and \(C \sin (\omega t+\phi)\) (a) By expanding \(\sin (\omega t+\phi)\) using the sum of the angles formula, show that the two expressions are equivalent if \(A=C \cos \phi\) and \(B=C \sin \phi\) (b) Consequently, show that \(A^{2}+B^{2}=C^{2}\) and that \(\phi\) then satisfies the equation \(\tan \phi=\frac{B}{A}\). (c) Generalize your result to state a proposition about \(A_{1} \sin \left(\omega t+\phi_{1}\right)+A_{2} \sin \left(\omega t+\phi_{2}\right)+A_{3} \sin \left(\omega t+\phi_{3}\right)\) (d) Write an essay, in your own words, that expresses the importance of the identity between \(A \sin (\omega t)+B \cos (\omega t)\) and \(C \sin (\omega t+\phi) .\) Be sure to note that \(|C| \geq \max (|A|,|B|)\) and that the identity holds only when you are forming a linear combination (adding and/or subtracting multiples of single powers) of sine and cosine of the same frequency.

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(y=x^{3}-x\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.