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

1. 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.
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.
3. 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.
4. 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.
5. 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.

Key Concepts

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.