Determine Which Of The Given Points Are On The Graph Of The Equation.Equation: Y = X 2 − X Y = X^2 - \sqrt{x} Y = X 2 − X Points: (0,0), (1,4), (-1,1)Which Of These Points Are On The Graph Of The Equation? Select All That Apply.A. (1,4) B. (0,0) C. (-1,1) D. None

Mar 13, 2025 by ADMIN 269 views

**Determine which of the given points are on the graph of the equation**

Introduction

In mathematics, graphing equations is a crucial concept that helps us visualize the relationship between variables. Given an equation, we can determine which points lie on its graph by substituting the x-coordinate of each point into the equation and checking if the resulting y-coordinate matches the given point. In this article, we will explore how to determine which of the given points are on the graph of the equation $y = x^2 - \sqrt{x}$ .

Understanding the Equation

The given equation is $y = x^2 - \sqrt{x}$ . To understand this equation, let's break it down into its components. The equation consists of two terms: $x^2$ and $-\sqrt{x}$ . The first term, $x^2$ , represents the square of the variable x. The second term, $-\sqrt{x}$ , represents the negative square root of x.

Substituting Points into the Equation

To determine which points lie on the graph of the equation, we need to substitute the x-coordinate of each point into the equation and check if the resulting y-coordinate matches the given point. Let's substitute each point into the equation:

Point (0,0)

To check if the point (0,0) lies on the graph of the equation, we substitute x = 0 into the equation:

$y = (0)^2 - \sqrt{0}$ $y = 0 - 0$ $y = 0$

Since the resulting y-coordinate matches the given point (0,0), we can conclude that the point (0,0) lies on the graph of the equation.

Point (1,4)

To check if the point (1,4) lies on the graph of the equation, we substitute x = 1 into the equation:

$y = (1)^2 - \sqrt{1}$ $y = 1 - 1$ $y = 0$

However, the resulting y-coordinate does not match the given point (1,4). Therefore, we can conclude that the point (1,4) does not lie on the graph of the equation.

Point (-1,1)

To check if the point (-1,1) lies on the graph of the equation, we substitute x = -1 into the equation:

$y = (-1)^2 - \sqrt{-1}$ $y = 1 - i$

However, the resulting y-coordinate is a complex number, and the point (-1,1) is a real number. Therefore, we can conclude that the point (-1,1) does not lie on the graph of the equation.

Conclusion

In conclusion, only the point (0,0) lies on the graph of the equation $y = x^2 - \sqrt{x}$ . The points (1,4) and (-1,1) do not lie on the graph of the equation.

Discussion

This problem requires a deep understanding of algebraic equations and graphing concepts. The equation $y = x^2 - \sqrt{x}$ is a quadratic equation with a square root term. To determine which points lie on the graph of the equation, we need to substitute the x-coordinate of each point into the equation and check if the resulting y-coordinate matches the given point.

Key Takeaways

To determine which points lie on the graph of an equation, we need to substitute the x-coordinate of each point into the equation and check if the resulting y-coordinate matches the given point.
The equation $y = x^2 - \sqrt{x}$ is a quadratic equation with a square root term.
Only the point (0,0) lies on the graph of the equation $y = x^2 - \sqrt{x}$ .

References

[1] Algebraic Equations and Graphing Concepts. (n.d.). Retrieved from https://www.mathopenref.com/graphing.html
[2] Quadratic Equations. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/quadratic-equations.html

Additional Resources

Khan Academy: Graphing Quadratic Equations
Mathway: Graphing Quadratic Equations
Wolfram Alpha: Graphing Quadratic Equations
Determine which of the given points are on the graph of the equation ===========================================================

Q&A: Determining Points on the Graph of an Equation

Q: What is the first step in determining which points lie on the graph of an equation? A: The first step is to substitute the x-coordinate of each point into the equation and check if the resulting y-coordinate matches the given point.

Q: How do I handle square root terms in an equation? A: When dealing with square root terms in an equation, you need to consider the domain of the square root function. The square root of a negative number is an imaginary number, so you need to be careful when substituting values into the equation.

Q: What is the difference between a quadratic equation and a linear equation? A: A quadratic equation is a polynomial equation of degree two, which means it has a squared variable term. A linear equation, on the other hand, is a polynomial equation of degree one, which means it has a variable term but no squared term.

Q: Can I use the same method to determine which points lie on the graph of a linear equation? A: Yes, you can use the same method to determine which points lie on the graph of a linear equation. However, since linear equations are simpler, you may be able to use a more straightforward approach, such as checking if the point lies on the line.

Q: What if I get a complex number as the result of substituting a point into the equation? A: If you get a complex number as the result of substituting a point into the equation, it means that the point does not lie on the graph of the equation. Complex numbers are used to represent numbers that have both real and imaginary parts.

Q: Can I use a graphing calculator to help me determine which points lie on the graph of an equation? A: Yes, you can use a graphing calculator to help you determine which points lie on the graph of an equation. Graphing calculators can graph equations and help you visualize the relationship between the variables.

Q: What if I'm not sure how to substitute a point into an equation? A: If you're not sure how to substitute a point into an equation, try breaking down the equation into its individual terms and substituting the point into each term separately. This can help you understand the equation better and make the substitution process easier.

Q: Can I use the same method to determine which points lie on the graph of a polynomial equation of degree three or higher? A: Yes, you can use the same method to determine which points lie on the graph of a polynomial equation of degree three or higher. However, since these equations are more complex, you may need to use more advanced techniques, such as factoring or using a graphing calculator.

Q: What if I get a negative value as the result of substituting a point into the equation? A: If you get a negative value as the result of substituting a point into the equation, it means that the point does not lie on the graph of the equation. However, be careful not to confuse a negative value with a complex number.

Q: Can I use the same method to determine which points lie on the graph of a rational equation? A: Yes, you can use the same method to determine which points lie on the graph of a rational equation. However, since rational equations have fractions, you need to be careful when substituting values into the equation.