In Exercises 47–50, 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

1. Step 1: Identify the variables in the sentence. Here, the variables are 'x' and 'y'.
2. Step 2: Translate the English sentence into a mathematical equation. The sentence 'The y-value is three decreased by the square of the x-value' can be written as: \( y = 3 - x^2 \).
3. Step 3: Recognize that this equation represents a parabola that opens downwards because the coefficient of \( x^2 \) is negative.
4. Step 4: Determine the vertex of the parabola. Since the equation is in the form \( y = a(x-h)^2 + k \), the vertex is at \( (0, 3) \).
5. Step 5: Graph the equation by plotting the vertex and additional points on either side of the vertex, then sketch the parabola.

Key Concepts

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 Quadratic Functions

Graphing quadratic functions involves plotting the relationship between the variables on a coordinate plane. The equation derived from the English sentence, y = 3 - x², represents a downward-opening parabola. Understanding how to graph such functions helps visualize the relationship between x and y, which is key to interpreting the behavior of the equation.