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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 29c
Chapter 1, Problem 29c

Graphing equations Graph the following equations. 


c. x² + 2x + y² + 4y + 1 = 0

Verified step by step guidance
1
Step 1: Recognize that the given equation is a conic section. The equation x^2 + 2x + y^2 + 4y + 1 = 0 is a quadratic equation in x and y, which suggests it could represent a circle, ellipse, parabola, or hyperbola.
Step 2: Complete the square for the x terms. Start with x^2 + 2x. To complete the square, take the coefficient of x, which is 2, divide it by 2 to get 1, and then square it to get 1. Add and subtract this square inside the equation: x^2 + 2x + 1 - 1.
Step 3: Complete the square for the y terms. Similarly, take the y terms y^2 + 4y. Take the coefficient of y, which is 4, divide it by 2 to get 2, and then square it to get 4. Add and subtract this square inside the equation: y^2 + 4y + 4 - 4.
Step 4: Rewrite the equation using the completed squares. The equation becomes (x + 1)^2 - 1 + (y + 2)^2 - 4 + 1 = 0. Simplify this to get (x + 1)^2 + (y + 2)^2 = 4.
Step 5: Identify the graph of the equation. The equation (x + 1)^2 + (y + 2)^2 = 4 is the equation of a circle with center at (-1, -2) and radius 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The equation given represents a conic section, specifically a circle or an ellipse, depending on the coefficients of the variables. Understanding how to identify and graph these shapes is essential for solving the problem.
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Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique simplifies the process of graphing by allowing us to rewrite the equation in vertex form, making it easier to identify the center and radius of the conic section.
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Graphing Techniques

Graphing techniques involve plotting points and understanding the shape of the graph based on the equation. For conic sections, knowing how to find key features such as vertices, axes of symmetry, and intercepts is crucial for accurately representing the graph on a coordinate plane.
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Related Practice
Textbook Question

Taxicab fees A taxicab ride costs \$3.50 plus \$2.50 per mile for the first 5 miles, with the rate dropping to \$1.50 per mile after the fifth mile. Let m be the distance (in miles) from the airport to a hotel. Find and graph the piecewise linear function c(m) that represents the cost of taking a taxi from the airport to a hotel m miles away.

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Textbook Question

Graphing functions Sketch a graph of each function.


ƒ(x) = { 2x if x ≤ 1 , 3-x if x > 1

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f(x)={3x−1, if x≤0−2x−1, if x>0f\(\left\)(x\(\right\))=\(\begin{cases}\)3x-1\(\frac{}{}\),\(\text{ if }\)x\(\le\)0\\ -2x-1,\(\text{ if }\)x>0\(\end{cases}\)

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Textbook Question

Find the inverse function (on the given interval, if specified) and graph both ff and f−1f^{-1} on the same set of axes. Check your work by looking for the required symmetry in the graphs.

f(x)=8−4xf\(\left\)(x\(\right\))=8-4x

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Textbook Question

The population of a small town was 500 in 2018 and is growing at a rate of 24 people per year. Find and graph the linear population function p(t) that gives the population of the town t years after 2018. Then use this model to predict the population in 2033.

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Textbook Question

{Use of Tech} Launching a rocket A small rocket is launched vertically upward from the edge of a cliff 8080 ft above the ground at a speed of 9696 ft/s. Its height (in feet) above the ground is given by h(t)=−16t2+96t+80h\(\left\)(t\(\right\))=-16t^2+96t+80, where tt represents time measured in seconds.

a. Assuming the rocket is launched at t=0t=0, what is an appropriate domain for hh?

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