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: 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: $y\; =\; 3\; -\; x^2$.
2. Step 2: Identify the two variables in the equation. Here, $x$ is the independent variable, and $y$ is the dependent variable.
3. 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 $y\; =\; 3\; -\; x^2$.
4. 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.
5. 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.

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 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.