Which Equation, When Graphed, Has $x$-intercepts At $(2,0)$ And $(4,0)$ And A $y$-intercept Of $(0,-16)$?A. $f(x)=-(x-2)(x-4)$B. $f(x)=-(x+2)(x+4)$C.
Introduction
When graphing equations, understanding the behavior of the function at specific points is crucial. In this case, we are given the x-intercepts at (2,0) and (4,0) and the y-intercept at (0,-16). We need to determine which equation, when graphed, will have these specific intercepts. In this article, we will explore the properties of quadratic functions and how to identify the correct equation.
Understanding Quadratic Functions
A quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic function is:
f(x) = ax^2 + bx + c
where a, b, and c are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve.
x-Intercepts and y-Intercept
The x-intercepts of a quadratic function are the points where the graph crosses the x-axis. At these points, the y-coordinate is zero. The y-intercept is the point where the graph crosses the y-axis. At this point, the x-coordinate is zero.
Given Information
We are given the x-intercepts at (2,0) and (4,0) and the y-intercept at (0,-16). This means that the graph of the function will cross the x-axis at x = 2 and x = 4, and it will cross the y-axis at y = -16.
Factoring the Equation
To find the equation of the quadratic function, we can use the fact that the x-intercepts are the roots of the equation. Since the x-intercepts are at x = 2 and x = 4, we can write the equation in factored form as:
f(x) = a(x - 2)(x - 4)
where a is a constant.
Determining the Value of a
We are also given the y-intercept at (0,-16). This means that when x = 0, y = -16. We can substitute these values into the equation to determine the value of a.
f(0) = -16 a(0 - 2)(0 - 4) = -16 a(-2)(-4) = -16 8a = -16 a = -2
Writing the Equation
Now that we have determined the value of a, we can write the equation of the quadratic function.
f(x) = -2(x - 2)(x - 4)
Checking the Answer Choices
We need to check if this equation matches any of the answer choices. Let's compare it to the answer choices:
A. f(x) = -(x - 2)(x - 4) B. f(x) = -(x + 2)(x + 4) C. f(x) = -(x - 2)(x - 4)
Our equation matches answer choice C.
Conclusion
In conclusion, the equation that, when graphed, has x-intercepts at (2,0) and (4,0) and a y-intercept of (0,-16) is:
f(x) = -(x - 2)(x - 4)
This equation matches answer choice C.
Discussion
This problem requires a good understanding of quadratic functions and their properties. It also requires the ability to analyze the given information and use it to determine the correct equation. The key concept in this problem is the fact that the x-intercepts are the roots of the equation, and the y-intercept can be used to determine the value of a.
Final Answer
The final answer is C.
Introduction
In our previous article, we explored the properties of quadratic functions and how to identify the correct equation given the x-intercepts and y-intercept. In this article, we will continue to delve deeper into the world of quadratic functions and answer some common questions related to x-intercepts and y-intercept.
Q&A
Q1: What are x-intercepts and y-intercept in a quadratic function?
A1: The x-intercepts of a quadratic function are the points where the graph crosses the x-axis. At these points, the y-coordinate is zero. The y-intercept is the point where the graph crosses the y-axis. At this point, the x-coordinate is zero.
Q2: How do I find the equation of a quadratic function given the x-intercepts and y-intercept?
A2: To find the equation of a quadratic function, you can use the fact that the x-intercepts are the roots of the equation. Since the x-intercepts are at x = 2 and x = 4, you can write the equation in factored form as:
f(x) = a(x - 2)(x - 4)
where a is a constant. You can then determine the value of a by substituting the y-intercept into the equation.
Q3: What is the relationship between the x-intercepts and the roots of the equation?
A3: The x-intercepts are the points where the graph crosses the x-axis, and they are also the roots of the equation. In other words, the x-intercepts are the values of x that make the equation equal to zero.
Q4: How do I determine the value of a in the equation?
A4: To determine the value of a, you can substitute the y-intercept into the equation. The y-intercept is the point where the graph crosses the y-axis, and it is given by the coordinates (0, y). You can substitute x = 0 and y = -16 into the equation to determine the value of a.
Q5: What is the significance of the y-intercept in a quadratic function?
A5: The y-intercept is the point where the graph crosses the y-axis, and it is given by the coordinates (0, y). The y-intercept is significant because it gives us information about the value of the function at x = 0.
Q6: Can I have multiple x-intercepts in a quadratic function?
A6: Yes, you can have multiple x-intercepts in a quadratic function. In fact, the number of x-intercepts is equal to the number of roots of the equation.
Q7: How do I graph a quadratic function given the x-intercepts and y-intercept?
A7: To graph a quadratic function, you can use the x-intercepts and y-intercept to determine the shape of the graph. The x-intercepts will give you the points where the graph crosses the x-axis, and the y-intercept will give you the point where the graph crosses the y-axis.
Conclusion
In conclusion, understanding x-intercepts and y-intercept is crucial in working with quadratic functions. By knowing how to find the equation of a quadratic function given the x-intercepts and y-intercept, you can analyze and graph quadratic functions with ease.
Final Answer
The final answer is that understanding x-intercepts and y-intercept is essential in working with quadratic functions.
Discussion
This article provides a comprehensive overview of x-intercepts and y-intercept in quadratic functions. It answers common questions related to these topics and provides a deeper understanding of the subject.
Additional Resources
For more information on quadratic functions, please refer to the following resources:
- Khan Academy: Quadratic Functions
- Mathway: Quadratic Functions
- Wolfram MathWorld: Quadratic Functions
Final Thoughts
Understanding x-intercepts and y-intercept is a fundamental concept in working with quadratic functions. By mastering this concept, you can analyze and graph quadratic functions with ease and solve problems related to these functions.