Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 49a

Chapter 2, Problem 49a

Write each English sentence as an equation in two variables. Then graph the equation. The y-value is three decreased by the square of the x-value.

Verified step by step guidance

Step 1: Start by translating the English sentence into a mathematical equation. The sentence states that the y-value is three decreased by the square of the x-value. This can be written as:

y = 3 - x^2

Step 2: Identify the two variables in the equation. Here,

x

is the independent variable, and

y

is the dependent variable.

Step 3: To graph the equation, create a table of values. Choose several values for

x

(e.g., -2, -1, 0, 1, 2) and calculate the corresponding

y

values using the equation

y = 3 - x^2

Step 4: Plot the points from the table of values on a coordinate plane. For example, if

x = -2

, calculate

y = 3 - (-2)^2 = 3 - 4 = -1

, so one point is (-2, -1). Repeat this for all chosen

x

values.

Step 5: Connect the plotted points with a smooth curve. Since the equation

y = 3 - x^2

represents a downward-opening parabola (due to the negative coefficient of

x^2

), the graph will have a vertex at the highest point (0, 3) and will curve downward symmetrically.

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Key Concepts

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

Variables in Equations

In algebra, variables are symbols that represent unknown values. In the context of the given question, 'x' and 'y' are the two variables used to form an equation. Understanding how to manipulate and interpret these variables is crucial for translating English sentences into mathematical equations.

Translating English Sentences to Equations

Translating English sentences into mathematical equations involves identifying the relationships described in the sentence. In this case, the phrase 'the y-value is three decreased by the square of the x-value' indicates a specific mathematical relationship that can be expressed as an equation: y = 3 - x². This skill is essential for solving problems in algebra.

Graphing Equations

Graphing equations involves plotting points on a coordinate plane to visually represent the relationship between the variables. For the equation y = 3 - x², the graph will be a downward-opening parabola. Understanding how to graph equations helps in analyzing their behavior and solutions, providing a visual interpretation of the mathematical relationships.

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Graphing Equations of Two Variables by Plotting Points

Related Practice

Textbook Question

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 1/(x - 1) + 5 = 11/(x - 1)

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Textbook Question

In Exercises 35–54, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? S = C/(1 - r) for r

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 5(x - 2) - 3(x + 4) ≥ 2x - 20

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Textbook Question

Solve each equation in Exercises 47–64 by completing the square. $x^{2} - 2 x = 2$

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Textbook Question

Solve each equation in Exercises 47–64 by completing the square. $x^{2} + 4 x = 12$

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