Ch. 1 - Equations and Inequalities

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

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 50a

Chapter 2, Problem 50a

Write each English sentence as an equation in two variables. Then graph the equation. The y-value is two more than 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 two more than the square of the x-value.' This can be written as y = x^2 + 2.

Step 2: To graph the equation, recognize that this is a quadratic equation in the form y = x^2 + c, where c = 2. This represents a parabola that opens upwards and is shifted 2 units up from the standard parabola y = x^2.

Step 3: Create a table of values for x and calculate the corresponding y-values using the equation y = x^2 + 2. For example, choose x-values such as -2, -1, 0, 1, and 2, and compute their y-values.

Step 4: Plot the points from the table of values on a coordinate plane. For instance, if x = -2, y = (-2)^2 + 2 = 6, so plot the point (-2, 6). Repeat this for all chosen x-values.

Step 5: Connect the plotted points with a smooth curve to complete the graph of the parabola. Ensure the curve reflects the symmetry of the parabola about the y-axis.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form y = ax^2 + bx + c. In this context, the relationship between the variables x and y involves squaring the x-value, which indicates that the graph will be a parabola. Understanding the properties of parabolas, such as their vertex and direction of opening, is essential for graphing the equation accurately.

Translating English Sentences to Equations

Translating English sentences into mathematical equations requires understanding the relationships described in the sentence. In this case, the phrase 'the y-value is two more than the square of the x-value' translates to the equation y = x^2 + 2. This skill is crucial for converting verbal descriptions into mathematical expressions that can be analyzed and graphed.

Graphing Equations

Graphing equations involves plotting points on a coordinate plane to visually represent the relationship between the variables. For the equation y = x^2 + 2, one would calculate y-values for various x-values and plot these points. Understanding how to interpret the graph, including identifying key features like intercepts and the shape of the curve, is vital for a complete analysis of the equation.

Recommended video:

Guided course

04:29

Graphing Equations of Two Variables by Plotting Points

Related Practice

Textbook Question

In Exercises 37–52, perform the indicated operations and write the result in standard form. (3√-5)(- 4√-12)

Textbook Question

Solve each equation in Exercises 47–64 by completing the square. $x^{2} - 6 x - 11 = 0$

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

views

Textbook Question

Perform the indicated operations and write the result in standard form. (7-i)(2+3i)

views

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. 3/(x + 4) - 7 = - 4/(x + 4)

views

Textbook Question

In Exercises 51–58, solve each compound inequality. 6 < x + 3 < 8

views