What Is The { Y $}$-intercept Of The Quadratic Function { F(x) = (x-8)(x+3) $}$?A. { (0, -24) $}$ B. { (-5, 0) $}$ C. { (0, 3) $}$ D. { (8, 0) $}$
Introduction
In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. One of the key concepts in quadratic functions is the y-intercept, which is the point where the graph of the function intersects the y-axis. In this article, we will explore the concept of the y-intercept in quadratic functions and how to find it.
What is the y-intercept?
The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. It is the value of the function when x is equal to zero. In other words, it is the value of the function at the point (0, y). The y-intercept is an important concept in mathematics, as it provides valuable information about the behavior of the function.
Finding the y-intercept of a Quadratic Function
To find the y-intercept of a quadratic function, we need to substitute x = 0 into the function and solve for y. This is because the y-intercept is the value of the function when x is equal to zero. Let's consider the quadratic function f(x) = (x-8)(x+3). To find the y-intercept, we need to substitute x = 0 into the function.
Step 1: Substitute x = 0 into the function
f(x) = (x-8)(x+3) f(0) = (0-8)(0+3) f(0) = (-8)(3) f(0) = -24
Step 2: Identify the y-intercept
The value of the function at x = 0 is -24, which means that the y-intercept of the quadratic function f(x) = (x-8)(x+3) is (0, -24).
Conclusion
In conclusion, the y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. It is the value of the function when x is equal to zero. To find the y-intercept of a quadratic function, we need to substitute x = 0 into the function and solve for y. In this article, we have explored the concept of the y-intercept in quadratic functions and how to find it.
Answer
The correct answer is A. (0, -24).
Additional Examples
Let's consider another example of a quadratic function: f(x) = (x+2)(x-5). To find the y-intercept, we need to substitute x = 0 into the function.
f(x) = (x+2)(x-5) f(0) = (0+2)(0-5) f(0) = (2)(-5) f(0) = -10
The y-intercept of the quadratic function f(x) = (x+2)(x-5) is (0, -10).
Tips and Tricks
- To find the y-intercept of a quadratic function, substitute x = 0 into the function and solve for y.
- The y-intercept is the value of the function at the point (0, y).
- The y-intercept provides valuable information about the behavior of the function.
Common Mistakes
- Not substituting x = 0 into the function to find the y-intercept.
- Not solving for y after substituting x = 0 into the function.
- Not identifying the y-intercept correctly.
Real-World Applications
The concept of the y-intercept in quadratic functions has many real-world applications. For example:
- In physics, the y-intercept of a quadratic function can represent the maximum height of an object.
- In economics, the y-intercept of a quadratic function can represent the minimum cost of production.
- In engineering, the y-intercept of a quadratic function can represent the maximum stress on a material.
Conclusion
Frequently Asked Questions
Q: What is the y-intercept of a quadratic function?
A: The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. It is the value of the function when x is equal to zero.
Q: How do I find the y-intercept of a quadratic function?
A: To find the y-intercept of a quadratic function, substitute x = 0 into the function and solve for y.
Q: What is the significance of the y-intercept in quadratic functions?
A: The y-intercept provides valuable information about the behavior of the function. It can represent the maximum height of an object, the minimum cost of production, or the maximum stress on a material.
Q: Can I find the y-intercept of a quadratic function using a graphing calculator?
A: Yes, you can find the y-intercept of a quadratic function using a graphing calculator. Simply graph the function and read the y-coordinate of the point where the graph intersects the y-axis.
Q: What is the difference between the y-intercept and the x-intercept of a quadratic function?
A: The y-intercept is the point where the graph of the function intersects the y-axis, while the x-intercept is the point where the graph of the function intersects the x-axis.
Q: Can I find the y-intercept of a quadratic function with a negative leading coefficient?
A: Yes, you can find the y-intercept of a quadratic function with a negative leading coefficient. Simply substitute x = 0 into the function and solve for y.
Q: What is the y-intercept of the quadratic function f(x) = (x-2)(x+4)?
A: To find the y-intercept of the quadratic function f(x) = (x-2)(x+4), substitute x = 0 into the function and solve for y.
f(x) = (x-2)(x+4) f(0) = (0-2)(0+4) f(0) = (-2)(4) f(0) = -8
The y-intercept of the quadratic function f(x) = (x-2)(x+4) is (0, -8).
Q: Can I use the y-intercept to determine the vertex of a quadratic function?
A: Yes, you can use the y-intercept to determine the vertex of a quadratic function. The vertex of a quadratic function is the point where the graph of the function changes direction.
Q: What is the y-intercept of the quadratic function f(x) = x^2 + 5x + 6?
A: To find the y-intercept of the quadratic function f(x) = x^2 + 5x + 6, substitute x = 0 into the function and solve for y.
f(x) = x^2 + 5x + 6 f(0) = (0)^2 + 5(0) + 6 f(0) = 6
The y-intercept of the quadratic function f(x) = x^2 + 5x + 6 is (0, 6).
Conclusion
In conclusion, the y-intercept of a quadratic function is an important concept in mathematics that provides valuable information about the behavior of the function. To find the y-intercept, substitute x = 0 into the function and solve for y. The y-intercept has many real-world applications and is an essential tool in various fields of study.