Question

Determine which of the given points are on the graph of the equation. $$ \begin{array}{l} \text { Equation: } y^{3}=x+1 \\ \text { Points: }(1,2) ;(0,1) ;(-1,0) \end{array} $$

Short answer

The points (0,1) and (-1,0) are on the graph.

Step-by-step solution

Substitute the first point into the equation

Take the point (1,2) and substitute x = 1 and y = 2 into the equation When substituting, calculate: \( y^3 = x + 1 \) \(2^3 = 1 + 1 \) \(8 = 2 \) Since 8 does not equal 2, point (1,2) is not on the graph of the equation.

Substitute the second point into the equation

Take the point (0,1) and substitute x = 0 and y = 1 into the equation When substituting, calculate: \( y^3 = x + 1 \) \(1^3 = 0 + 1 \) \(1 = 1 \) Since 1 equals 1, point (0,1) is on the graph of the equation.

Substitute the third point into the equation

Take the point (-1,0) and substitute x = -1 and y = 0 into the equation When substituting, calculate: \( y^3 = x + 1 \) \(0^3 = -1 + 1 \) \(0 = 0 \) Since 0 equals 0, point (-1,0) is on the graph of the equation.

Key concepts

Substitution Method

The substitution method is key to understanding how to determine if a point lies on a graph of an equation. This technique involves replacing variables in an equation with known values from a given point. For instance, in the equation provided, we use the coordinates of each point to see if they satisfy the equation.

• Start by identifying the x and y values of your point.
• Substitute these values into the original equation.
• Simplify the equation, performing any necessary calculations.
• Check if the resulting statement is true.

Following these steps will allow you to verify if a point lies on the curve defined by the equation.

Verifying Points

Verifying points is a crucial aspect of graphing equations. It helps confirm whether a given point fits the equation or not. In the provided problem, we have three points and a cubic equation:

After substituting each point into the equation:

• (1, 2) results in an incorrect statement (8 ≠ 2), hence it does not lie on the graph.
• (0, 1) gives a true statement (1 = 1), indicating it lies on the graph.
• (-1, 0) also gives a true statement (0 = 0), confirming it lies on the graph.

Verifying points through substitution is essential for correctly plotting points on a graph.

Cubic Equations

Understanding cubic equations is fundamental for graphing and solving problems like the one provided. A cubic equation is expressed as:

y^3 = x + 1

Characteristics of cubic equations include:

• The highest power of the variable (here, y) is 3.
• These equations can have one or more solutions for a given value of y.
• The graph of a cubic equation can have complex shapes, including curves and inflection points.

The provided equation is simpler, but it's a perfect example to illustrate how substituting and verifying points can confirm if they lie on the curve defined by the cubic equation. Understanding these characteristics helps in visualizing and plotting cubic equations precisely.