Find The \[$ Y \$\]-intercept Of The Equation \[$ Y = (x-3)^2 + 2 \$\].A. \[$(0, 2)\$\] B. \[$(0, -11)\$\] C. \[$(0, 11)\$\] D. \[$(0, -3)\$\]
Introduction
In mathematics, the y-intercept of a quadratic equation is the point at which the graph of the equation intersects the y-axis. This point is also known as the origin, and it is denoted by the coordinates (0, y). In this article, we will explore how to find the y-intercept of a quadratic equation, using the equation y = (x-3)^2 + 2 as an example.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (x) is two. The general form of a quadratic equation is:
y = ax^2 + bx + c
where a, b, and c are constants, and x is the variable. In the equation y = (x-3)^2 + 2, the term (x-3)^2 is a quadratic expression, and the constant term 2 is added to it.
Finding the y-Intercept
To find the y-intercept of a quadratic equation, we need to substitute x = 0 into the equation and solve for y. This is because the y-intercept is the point at which the graph of the equation intersects the y-axis, and the x-coordinate of this point is always 0.
Let's substitute x = 0 into the equation y = (x-3)^2 + 2:
y = (0-3)^2 + 2 y = (-3)^2 + 2 y = 9 + 2 y = 11
Therefore, the y-intercept of the equation y = (x-3)^2 + 2 is the point (0, 11).
Why is the y-Intercept Important?
The y-intercept is an important concept in mathematics, particularly in algebra and calculus. It is used to determine the behavior of a function, and it is also used to find the maximum or minimum value of a function.
In the context of the equation y = (x-3)^2 + 2, the y-intercept is the point at which the graph of the equation intersects the y-axis. This point is also the minimum value of the function, and it is the lowest point on the graph.
Conclusion
In conclusion, finding the y-intercept of a quadratic equation is a simple process that involves substituting x = 0 into the equation and solving for y. The y-intercept is an important concept in mathematics, and it is used to determine the behavior of a function and find the maximum or minimum value of a function.
Example Problems
Here are a few example problems to help you practice finding the y-intercept of a quadratic equation:
- Find the y-intercept of the equation y = (x+2)^2 - 3.
- Find the y-intercept of the equation y = (x-1)^2 + 4.
- Find the y-intercept of the equation y = (x+1)^2 - 2.
Solutions
- y = (0+2)^2 - 3 y = (2)^2 - 3 y = 4 - 3 y = 1
Therefore, the y-intercept of the equation y = (x+2)^2 - 3 is the point (0, 1).
- y = (0-1)^2 + 4 y = (-1)^2 + 4 y = 1 + 4 y = 5
Therefore, the y-intercept of the equation y = (x-1)^2 + 4 is the point (0, 5).
- y = (0+1)^2 - 2 y = (1)^2 - 2 y = 1 - 2 y = -1
Therefore, the y-intercept of the equation y = (x+1)^2 - 2 is the point (0, -1).
Final Answer
Q: What is the y-intercept of a quadratic equation?
A: The y-intercept of a quadratic equation is the point at which the graph of the equation intersects the y-axis. This point is also known as the origin, and it is denoted by the coordinates (0, y).
Q: How do I find the y-intercept of a quadratic equation?
A: To find the y-intercept of a quadratic equation, you need to substitute x = 0 into the equation and solve for y. This is because the y-intercept is the point at which the graph of the equation intersects the y-axis, and the x-coordinate of this point is always 0.
Q: What is the general form of a quadratic equation?
A: The general form of a quadratic equation is:
y = ax^2 + bx + c
where a, b, and c are constants, and x is the variable.
Q: Can you give an example of a quadratic equation?
A: Yes, here is an example of a quadratic equation:
y = (x-3)^2 + 2
Q: How do I find the y-intercept of the equation y = (x-3)^2 + 2?
A: To find the y-intercept of the equation y = (x-3)^2 + 2, you need to substitute x = 0 into the equation and solve for y. Here's how you do it:
y = (0-3)^2 + 2 y = (-3)^2 + 2 y = 9 + 2 y = 11
Therefore, the y-intercept of the equation y = (x-3)^2 + 2 is the point (0, 11).
Q: What is the significance of the y-intercept in a quadratic equation?
A: The y-intercept is an important concept in mathematics, particularly in algebra and calculus. It is used to determine the behavior of a function, and it is also used to find the maximum or minimum value of a function.
Q: Can you give an example of a quadratic equation with a negative y-intercept?
A: Yes, here is an example of a quadratic equation with a negative y-intercept:
y = (x+2)^2 - 3
To find the y-intercept of this equation, you need to substitute x = 0 into the equation and solve for y. Here's how you do it:
y = (0+2)^2 - 3 y = (2)^2 - 3 y = 4 - 3 y = 1
Therefore, the y-intercept of the equation y = (x+2)^2 - 3 is the point (0, 1).
Q: Can you give an example of a quadratic equation with a positive y-intercept?
A: Yes, here is an example of a quadratic equation with a positive y-intercept:
y = (x-1)^2 + 4
To find the y-intercept of this equation, you need to substitute x = 0 into the equation and solve for y. Here's how you do it:
y = (0-1)^2 + 4 y = (-1)^2 + 4 y = 1 + 4 y = 5
Therefore, the y-intercept of the equation y = (x-1)^2 + 4 is the point (0, 5).
Q: Can you give an example of a quadratic equation with a zero y-intercept?
A: Yes, here is an example of a quadratic equation with a zero y-intercept:
y = (x+1)^2 - 2
To find the y-intercept of this equation, you need to substitute x = 0 into the equation and solve for y. Here's how you do it:
y = (0+1)^2 - 2 y = (1)^2 - 2 y = 1 - 2 y = -1
Therefore, the y-intercept of the equation y = (x+1)^2 - 2 is the point (0, -1).
Q: How do I determine the behavior of a function using the y-intercept?
A: The y-intercept can be used to determine the behavior of a function. If the y-intercept is positive, the function is increasing. If the y-intercept is negative, the function is decreasing. If the y-intercept is zero, the function is constant.
Q: Can you give an example of how to use the y-intercept to determine the behavior of a function?
A: Yes, here is an example of how to use the y-intercept to determine the behavior of a function:
Suppose we have the function y = (x-3)^2 + 2. To determine the behavior of this function, we need to find the y-intercept. We have already done this in a previous example, and we found that the y-intercept is the point (0, 11).
Since the y-intercept is positive, we know that the function is increasing. This means that as x increases, y also increases.
Conclusion
In conclusion, the y-intercept is an important concept in mathematics, particularly in algebra and calculus. It is used to determine the behavior of a function, and it is also used to find the maximum or minimum value of a function. By understanding how to find the y-intercept of a quadratic equation, you can better understand the behavior of a function and make more informed decisions in your mathematical calculations.