/*! 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 74 Graph each relation on a graphin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 74

Graph each relation on a graphing calculator by solving for \(y\) and graphing two functions. $$x=4 y^{2}+4 y+1$$

Short Answer

Expert verified
The functions to graph are \(y_1 = \frac{-1 + \sqrt{x}}{2}\) and \(y_2 = \frac{-1 - \sqrt{x}}{2}\).

Step by step solution

01

Rewrite the Equation

Rewrite the equation in a standard quadratic form in terms of y. The given equation is \[ x = 4y^2 + 4y + 1\]
02

Solve for y

To graph the relation, we need to solve for y. Rewrite the equation in the form of a quadratic equation in terms of y:\[ 4y^2 + 4y + 1 - x = 0 \]This is a standard quadratic equation of the form \[ ay^2 + by + c = 0 \]where \[ a = 4, b = 4, \text{ and } c = 1 - x \]
03

Apply the Quadratic Formula

The quadratic formula is given by \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).Substitute the values of a, b, and c into the quadratic formula:\[ y = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 4 \cdot (1 - x)}}{2 \cdot 4} = \frac{-4 \pm \sqrt{16 - 16 + 16x}}{8} = \frac{-4 \pm \sqrt{16x}}{8} \]Further simplify to find the solutions for y:\[ y = \frac{-4 \pm 4\sqrt{x}}{8} = \frac{-4 (1 \pm \sqrt{x})}{8} = \frac{-1 \pm \sqrt{x}}{2} \]
04

Express y in Terms of x

The solutions to y in terms of x are:\[ y = \frac{-1 + \sqrt{x}}{2} \]and\[ y = \frac{-1 - \sqrt{x}}{2} \]These represent two functions: \[ y_1 = \frac{-1 + \sqrt{x}}{2} \]and\[ y_2 = \frac{-1 - \sqrt{x}}{2} \]
05

Graph Each Function

Graph the two functions obtained from solving for y. Enter these equations into a graphing calculator:\( y_1 = \frac{-1 + \sqrt{x}}{2} \)and\( y_2 = \frac{-1 - \sqrt{x}}{2} \).These will give the graph of the original relation, and the resulting graph will show both branches of the curve.

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.

Quadratic Equation
A quadratic equation is a polynomial equation of the second degree. It takes the general form \[ ax^2 + bx + c = 0 \] where * \(a, b,\) and \(c\) are constants, and * \(x\) is the variable. In this particular exercise, we have the equation \[ 4y^2 + 4y + 1 - x = 0 \] where * \(a = 4\), * \(b = 4\), and * \(c = 1 - x\).
Understanding how to manipulate and solve quadratic equations is crucial for graphing quadratic relations.
Graphing Calculator
A graphing calculator is a powerful tool that helps you visualize mathematical functions. It allows you to input equations and see their graphs on a screen. For this exercise, you will input the two functions\( y_1 = \frac{-1 + \sqrt{x}}{2} \) and \( y_2 = \frac{-1 - \sqrt{x}}{2} \).
Helpful tips on using a graphing calculator:
  • Make sure you enter the equations correctly.
  • Check your graph window settings to ensure it covers the relevant range for \(x\) and \(y\).
  • Use the zoom features to get a clearer view of the graph.
Graphing these functions will give you the visual representation of the relation \( x = 4y^2 + 4y + 1 \).
Quadratic Formula
The quadratic formula is a key tool for solving quadratic equations. It is given by: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where:
  • \(a\) is the coefficient of \(y^2\)
  • \(b\) is the coefficient of \(y\)
  • \(c\) is the constant term
In this exercise, you use the quadratic formula to solve for \(y\) after rewriting the original equation as \[ 4y^2 + 4y + 1 - x = 0 \] It's essential to substitute the correct values for \(a\), \(b\), and \(c\) to get the accurate solutions for \(y\). Solving it, we get: \[ y = \frac{-4 \pm \sqrt{16x}}{8} \] Further simplification leads to the two functions: \[ y_1 = \frac{-1 + \sqrt{x}}{2} \] and \[ y_2 = \frac{-1 - \sqrt{x}}{2} \]
These solutions express \(y\) in terms of \(x\), making it easier to graph the relations.
Solving for y
Solving for \(y\) means isolating \(y\) on one side of the equation to express it as a function of \(x\). Here's how it was done step-by-step in this exercise:
  • First, rewrite the equation \[ x = 4y^2 + 4y + 1\] in a standard quadratic form: \[ 4y^2 + 4y + 1 - x = 0\]
  • Next, use the quadratic formula \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] with \(a = 4\), \(b = 4\), and \(c = 1 - x\).
  • Simplify the solutions to provide the two functions: \[ y_1 = \frac{-1 + \sqrt{x}}{2}\] and \[ y_2 = \frac{-1 - \sqrt{x}}{2}\]
By solving for \(y\), you end up with two different functions which you can then graph separately using a graphing calculator. This helps in visualizing the entire relation.

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

Graph both equations of each system on the same coordinate axes. Use elimination of variables to find all points of intersection. $$ \begin{aligned} &x^{2}+y^{2}=25\\\ &x^{2}+25 y^{2}=25 \end{aligned} $$

Sketch the graph of each parabola. $$ y=(x-2)^{2}+3 $$

$$\text {Graph each equation.}$$ $$y=\sqrt{1-x^{2}}$$

Sketch the graph of each ellipse. $$ \frac{(x+1)^{2}}{16}+\frac{(y-2)^{2}}{25}=1 $$

Graph both equations of each system on the same coordinate axes. Use elimination of variables to find all points of intersection. $$ \begin{aligned} &y=x^{2}+5 x+6\\\ &y=x+11 \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.