List The Intercepts For The Equation $y^2 = X + 36$.Select The Correct Choice Below And Fill In Any Answer Boxes Within Your Choice.A. The Intercept(s) Is/are $\square$.B. There Are No Intercepts.

Feb 26, 2025 by ADMIN 197 views

**Solving the Equation $y^2 = x + 36$ for Intercepts**

Introduction

In mathematics, an intercept is a point where a line or a curve intersects the x-axis or the y-axis. In this article, we will focus on finding the intercepts for the equation $y^2 = x + 36$ . This equation represents a parabola, and we will use algebraic methods to find its intercepts.

Understanding the Equation

The given equation is $y^2 = x + 36$ . To find the intercepts, we need to understand that the x-intercept occurs when $y = 0$ , and the y-intercept occurs when $x = 0$ . We will use these conditions to find the intercepts.

Finding the X-Intercept

To find the x-intercept, we set $y = 0$ in the equation $y^2 = x + 36$ . This gives us:

0^2 = x + 36

Simplifying the equation, we get:

0 = x + 36

Subtracting 36 from both sides, we get:

-36 = x

Therefore, the x-intercept is $x = -36$ .

Finding the Y-Intercept

To find the y-intercept, we set $x = 0$ in the equation $y^2 = x + 36$ . This gives us:

y^2 = 0 + 36

Simplifying the equation, we get:

y^2 = 36

Taking the square root of both sides, we get:

y = \pm \sqrt{36}

Simplifying further, we get:

y = \pm 6

Therefore, the y-intercepts are $y = 6$ and $y = -6$ .

Conclusion

In this article, we have found the intercepts for the equation $y^2 = x + 36$ . The x-intercept is $x = -36$ , and the y-intercepts are $y = 6$ and $y = -6$ . We have used algebraic methods to find these intercepts, and we have understood the conditions for finding x-intercepts and y-intercepts.

Answer

The correct answer is:

A. The intercept(s) is/are $\boxed{-36, 6, -6}$ .

Discussion

The equation $y^2 = x + 36$ represents a parabola that opens to the right. The x-intercept is the point where the parabola intersects the x-axis, and the y-intercepts are the points where the parabola intersects the y-axis. We have found these intercepts using algebraic methods, and we have understood the conditions for finding x-intercepts and y-intercepts.

References

Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships. We can use algebraic methods to solve equations, graph functions, and find intercepts.
Geometry: Geometry is a branch of mathematics that deals with the study of shapes and their properties. We can use geometric methods to graph parabolas, find intercepts, and solve problems involving shapes.
Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and accumulation. We can use calculus to find the derivatives and integrals of functions, and we can use these concepts to solve problems involving rates of change and accumulation.
Frequently Asked Questions (FAQs) about Intercepts =====================================================

Q: What is an intercept in mathematics?

A: An intercept is a point where a line or a curve intersects the x-axis or the y-axis. In other words, it is a point where the graph of a function touches the x-axis or the y-axis.

Q: How do I find the x-intercept of a function?

A: To find the x-intercept of a function, set the y-coordinate to 0 and solve for the x-coordinate. This means that you need to substitute 0 for y in the equation of the function and solve for x.

Q: How do I find the y-intercept of a function?

A: To find the y-intercept of a function, set the x-coordinate to 0 and solve for the y-coordinate. This means that you need to substitute 0 for x in the equation of the function and solve for y.

Q: What is the difference between an x-intercept and a y-intercept?

A: The main difference between an x-intercept and a y-intercept is the axis that the function intersects. An x-intercept is the point where the function intersects the x-axis, while a y-intercept is the point where the function intersects the y-axis.

Q: Can a function have more than one intercept?

A: Yes, a function can have more than one intercept. For example, a quadratic function can have two x-intercepts and two y-intercepts.

Q: How do I graph a function with intercepts?

A: To graph a function with intercepts, start by plotting the intercepts on the graph. Then, use the equation of the function to determine the shape of the graph. Finally, plot additional points on the graph to complete it.

Q: What are some common types of functions that have intercepts?

A: Some common types of functions that have intercepts include linear functions, quadratic functions, and polynomial functions.

Q: Can I use technology to find intercepts?

A: Yes, you can use technology such as graphing calculators or computer software to find intercepts. These tools can help you graph functions and find their intercepts quickly and accurately.

Q: What are some real-world applications of intercepts?

A: Intercepts have many real-world applications, including physics, engineering, and economics. For example, in physics, intercepts can be used to model the motion of objects, while in engineering, intercepts can be used to design buildings and bridges.

Q: How do I use intercepts to solve problems?

A: To use intercepts to solve problems, start by identifying the type of function that is involved. Then, use the equation of the function to find the intercepts. Finally, use the intercepts to solve the problem.

Q: What are some common mistakes to avoid when finding intercepts?

A: Some common mistakes to avoid when finding intercepts include:

Not setting the correct variable to 0
Not solving the equation correctly
Not plotting the intercepts correctly on the graph
Not using the correct equation of the function

By avoiding these mistakes, you can ensure that you find the correct intercepts for a function.