Textbook Question
In Exercises 11–26, determine whether each equation defines y as a function of x. 4x = y²
Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 3, Problem 18
Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (-1/4, -1/7) and (3/4, 6/7)
Verified step by step guidance1
Identify the coordinates of the two points: \(\left(-\frac{1}{4}, -\frac{1}{7}\right)\) and \(\left(\frac{3}{4}, \frac{6}{7}\right)\).
Recall the distance formula between two points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\):
\[d = \sqrt{\left(x_2 - x_1\right)^2 + \left(y_2 - y_1\right)^2}\]
Calculate the difference in the \(x\)-coordinates:
\[x_2 - x_1 = \frac{3}{4} - \left(-\frac{1}{4}\right) = \frac{3}{4} + \frac{1}{4}\]
Calculate the difference in the \(y\)-coordinates:
\[y_2 - y_1 = \frac{6}{7} - \left(-\frac{1}{7}\right) = \frac{6}{7} + \frac{1}{7}\]
Substitute these differences into the distance formula, simplify the expression under the square root to get the simplified radical form, and then approximate the decimal value rounded to two decimal places.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Distance Formula
The distance formula calculates the length between two points in a coordinate plane. It is derived from the Pythagorean theorem and given by √[(x2 - x1)² + (y2 - y1)²]. This formula helps find the straight-line distance between points with coordinates (x1, y1) and (x2, y2).
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Simplifying Radical Expressions
Simplifying radicals involves reducing the square root expressions to their simplest form by factoring out perfect squares. This process makes the answer more exact and easier to interpret before rounding. For example, √50 can be simplified to 5√2.
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Rounding Decimal Numbers
Rounding is the process of approximating a number to a specified number of decimal places for clarity and practicality. After calculating the exact distance, rounding to two decimal places provides a concise and usable numerical answer, such as 1.23 instead of 1.23456.
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Related Practice
Textbook Question
In Exercises 11–26, determine whether each equation defines y as a function of x. x = y²
Textbook Question
Use the graph to determine (a) the function's domain, (b) the function's range, (c) the x-intercepts, if any, (d) the y-intercept, if there is one, (e) intervals on which the function is increasing, decreasing or constant, (f) the missing function values, indicated by question marks, below each graph.
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