For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. y = -x³

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Identify the given equation: \(y = -x^2\). This is a quadratic function where the parabola opens downward because of the negative sign in front of \(x^2\).

Choose at least three values for \(x\) to find corresponding \(y\) values. For example, select \(x = -1\), \(x = 0\), and \(x = 1\).

Calculate the \(y\) values using the equation for each chosen \(x\): For \(x = -1\), compute \(y = -(-1)^2\); For \(x = 0\), compute \(y = -(0)^2\); For \(x = 1\), compute \(y = -(1)^2\).

Create a table of ordered pairs \((x, y)\) using the values found in the previous step. This table will help visualize points on the graph.

Use the table of points to plot the graph of the equation on the coordinate plane. Remember, the graph is a downward-opening parabola with its vertex at the origin \((0,0)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Understanding Quadratic Functions

A quadratic function is a polynomial of degree two, typically written as y = ax^2 + bx + c. In this case, y = -x^2 is a simple quadratic with a negative leading coefficient, which means its graph is a parabola opening downward. Recognizing the shape and properties of quadratic functions helps in plotting points and understanding their behavior.

Creating a Table of Ordered Pairs

To find ordered pairs that satisfy the equation, select values for x and compute the corresponding y values. For y = -x^2, plug in different x values (e.g., -1, 0, 1) to get y values (-1, 0, -1). This table of points provides specific solutions that can be plotted on a coordinate plane.

Graphing the Equation

Graphing involves plotting the ordered pairs on the coordinate plane and connecting them smoothly to reveal the shape of the function. For y = -x^2, the points form a downward-opening parabola symmetric about the y-axis. Understanding symmetry and vertex location aids in sketching an accurate graph.

For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. y = -x3