Question

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

Short answer

The points (0,0) and (1,-1) are on the graph. The point (1,1) is not.

Step-by-step solution

Evaluate the equation at the first point

Insert the coordinates of the first point \(0,0\) into the equation \[ y=x^{3}-2\sqrt{x} \]. This gives us \(y = 0^3 - 2\sqrt{0} = 0 \). Since \(y\) here is 0, which matches the given \(y\)-coordinate (0), the point \( (0,0) \) is on the graph.

Evaluate the equation at the second point

Insert the coordinates of the second point \(1,1\) into the equation \[ y=x^{3}-2\sqrt{x} \]. This gives us \(y = 1^3 - 2\sqrt{1} = 1 - 2 = -1 \). Since \(y\) here is -1, and this is not equivalent to the given \(y\)-coordinate (1), the point \( (1,1) \) is not on the graph.

Evaluate the equation at the third point

Insert the coordinates of the third point \(1,-1\) into the equation \[ y=x^{3}-2\sqrt{x} \]. This gives us \(y = 1^3 - 2\sqrt{1} = 1 - 2 = -1 \). Since \(y\) here is -1, which matches the given \(y\)-coordinate (-1), the point \( (1,-1) \) is on the graph.

Key concepts

Evaluating Functions

To understand whether a point lies on the graph of a given equation, we need to evaluate the function at the given x-coordinates and see if the y-coordinate matches the result. Evaluating functions means substituting the x-values from the coordinate points into the equation and solving for y. For instance, given the equation \( y = x^3 - 2\sqrt{x} \), here’s how you evaluate it for different points:

• Start by plugging in the x-value of the point. For instance, if x = 0 for the point (0,0), then the equation becomes \( y = 0^3 - 2\sqrt{0} = 0 \).
• Next, compute the right-hand side to find y. In our example, y equals 0, which matches the y-value of the point.

Thus, (0,0) is on the graph. By evaluating the function like this for all points, you can verify if they lie on the graph or not.

Coordinate Points

Coordinate points are a pair of numbers (x, y) that define a specific location on a graph. The first number represents the position along the x-axis and the second number represents the position along the y-axis.
When working with coordinate points:

• First, identify the x-coordinate (the horizontal placement).
• Second, identify the y-coordinate (the vertical placement).
• Substitute the x-coordinate into your given equation to solve for y.

In the given problem, checking if points (0,0), (1,1), and (1,-1) fall on the graph involves using each x-coordinate within the function and comparing the result with the corresponding y-coordinate.

Polynomials

Polynomials are algebraic expressions that include terms in the form of \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer exponent. The given equation \( y = x^3 - 2\sqrt{x} \) is a combination of a polynomial term (\