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Chapter 1: Problem 45

Find an expression for the function whose graph is the given curve. The bottom half of the parabola \(x+(y-1)^{2}=0\)

Short Answer

Expert verified
The function is \( y = 1 - \sqrt{-x} \).

Step by step solution

01

Identify the given equation

We start by examining the given equation, which is a parabola: \[ x + (y - 1)^2 = 0 \]This equation represents a parabola that opens horizontally rather than vertically.
02

Solve for y

To express the given equation as a function of \(x\), we solve for \(y\):\[ (y - 1)^2 = -x \] Taking the square root of both sides, we obtain:\[ y - 1 = \pm \sqrt{-x} \]Thus, \[ y = 1 \pm \sqrt{-x} \]
03

Determine the correct branch

Since we are tasked with finding the bottom half of the parabola, we select the negative branch of the square root:\[ y = 1 - \sqrt{-x} \] This function represents the bottom half of the parabola because it decreases as \(x\) moves leftward (decreasing).
04

Verify the function

Substitute back into the original equation to verify: \[ x + ((1 - \sqrt{-x}) - 1)^2 = 0 \] Simplifying, \[ x + (-\sqrt{-x})^2 = 0 \] \[ x + (-x) = 0 \] which is true. Thus, the function \(y = 1 - \sqrt{-x}\) correctly describes the bottom half of the parabola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exploring the Function of x
When we talk about a function of x, we are referring to a type of mathematical relationship. This relationship involves an equation where the value of the variable y is determined based on the input x.
  • An equation like this, characterized by y being expressed in terms of x, is called a function.
  • The notation often used for functions is f(x), meaning the function dependent on x, or it can simply be written as y.
In the problem we have, the equation \(x + (y - 1)^2 = 0\) needs to be expressed as a function of x. For this, we isolate y on one side. After manipulation, we achieve the expression \(y = 1 - \sqrt{-x}\), which defines the bottom half of the parabola.
Understanding that we have a function of x helps us recognize which y-coordinates the parabola will cover as x values change.
Mastering Solving Quadratic Equations
Solving quadratic equations is a fundamental skill in algebra, useful for various applications such as finding the roots of a polynomial or expressing a parabola in standard terms.
Quadratic equations generally have the form \(ax^2 + bx + c = 0\). In this exercise though, we encounter an unconventional parabola equation, \(x + (y - 1)^2 = 0\).
  • To solve this, especially for a function of one variable, we perform algebraic manipulations to isolate the variable of interest, y in this case.
  • This involves rearranging terms and, as evidenced here, sometimes taking square roots to simplify to \(y = 1 \pm \sqrt{-x}\).
Quadratic equations typically offer two solutions, represented by the \(\pm\) sign. For this problem, and specifically targeting the bottom half of the parabola, we select the negative root, leading to \(y = 1 - \sqrt{-x}\). This manipulation allows us to model the specific curve section we're interested in.
Understanding Horizontal Parabolas
Typically, parabolas are vertical, opening up or down along the y-axis. However, horizontal parabolas flip this standard orientation, opening left or right along the x-axis.
The equation \(x + (y - 1)^2 = 0\) describes such a horizontal parabola. Here, instead of x being dependent on y, the role is reversed, making the traditional vertex form more focused on the y variable.
  • Horizontal parabolas are indicated by the squared term being associated with y (as in \((y-1)^2\)), unlike vertical ones where x is squared.
  • In this exercise, the equation shows the parabola opening to the left rather than the right. This is due to a negative x term resulting from solving \((y - 1)^2 = -x\).
Our specific interest here is in the bottom half of this horizontal parabola, modeled by the function \(y = 1 - \sqrt{-x}\). This function reflects how y decreases as x becomes more negative, adhering to the parabolic shape and orientation.

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