The Graph Of The Function $f(x) = X^2 - 4x + 6$ Is Shown Here. What Is Its Axis Of Symmetry?A. $x = 0$ B. \$x = 2$[/tex\] C. $x = 6$ D. $x = -2$
Introduction
When analyzing the graph of a quadratic function, one of the key concepts to understand is the axis of symmetry. This concept is crucial in identifying the shape and behavior of the graph. In this article, we will explore the axis of symmetry of the quadratic function $f(x) = x^2 - 4x + 6$ and discuss its significance.
What is the Axis of Symmetry?
The axis of symmetry is a vertical line that passes through the vertex of a parabola. It is a line of symmetry that divides the parabola into two equal halves. The axis of symmetry is denoted by the equation $x = a$, where $a$ is the x-coordinate of the vertex.
Finding the Axis of Symmetry
To find the axis of symmetry of a quadratic function, we can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. In the case of the function $f(x) = x^2 - 4x + 6$, we have $a = 1$ and $b = -4$.
Calculating the Axis of Symmetry
Using the formula $x = -\frac{b}{2a}$, we can calculate the axis of symmetry as follows:
Conclusion
Therefore, the axis of symmetry of the quadratic function $f(x) = x^2 - 4x + 6$ is $x = 2$. This means that the graph of the function is symmetric about the vertical line $x = 2$.
Significance of the Axis of Symmetry
The axis of symmetry is an important concept in understanding the behavior of a quadratic function. It helps us identify the vertex of the parabola, which is the highest or lowest point of the graph. The axis of symmetry also helps us determine the direction of the parabola's opening, which is either upwards or downwards.
Real-World Applications
The axis of symmetry has numerous real-world applications in various fields, including physics, engineering, and economics. For example, in physics, the axis of symmetry is used to describe the motion of objects under the influence of gravity. In engineering, the axis of symmetry is used to design structures that are resistant to stress and strain. In economics, the axis of symmetry is used to model the behavior of economic systems and make predictions about future trends.
Examples of Quadratic Functions with Different Axes of Symmetry
Here are some examples of quadratic functions with different axes of symmetry:
-
f(x) = x^2 + 4x + 6$ has an axis of symmetry of $x = -2$.
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f(x) = x^2 - 2x + 6$ has an axis of symmetry of $x = 1$.
-
f(x) = x^2 + 6x + 6$ has an axis of symmetry of $x = -3$.
Conclusion
In conclusion, the axis of symmetry is a crucial concept in understanding the graph of a quadratic function. It helps us identify the vertex of the parabola and determine the direction of the parabola's opening. The axis of symmetry has numerous real-world applications in various fields, including physics, engineering, and economics. By understanding the axis of symmetry, we can better analyze and model the behavior of quadratic functions.
Final Answer
The final answer is:
Introduction
In our previous article, we explored the concept of the axis of symmetry of a quadratic function. In this article, we will provide a Q&A guide to help you better understand the graph of a quadratic function and its axis of symmetry.
Q: What is the axis of symmetry of a quadratic function?
A: The axis of symmetry is a vertical line that passes through the vertex of a parabola. It is a line of symmetry that divides the parabola into two equal halves.
Q: How do I find the axis of symmetry of a quadratic function?
A: To find the axis of symmetry of a quadratic function, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.
Q: What is the significance of the axis of symmetry?
A: The axis of symmetry is an important concept in understanding the behavior of a quadratic function. It helps us identify the vertex of the parabola, which is the highest or lowest point of the graph. The axis of symmetry also helps us determine the direction of the parabola's opening, which is either upwards or downwards.
Q: How do I determine the direction of the parabola's opening?
A: To determine the direction of the parabola's opening, you can look at the coefficient of the $x^2$ term. If the coefficient is positive, the parabola opens upwards. If the coefficient is negative, the parabola opens downwards.
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the highest or lowest point of the graph. It is the point where the parabola changes direction.
Q: How do I find the vertex of a parabola?
A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. The y-coordinate of the vertex can be found by plugging the x-coordinate into the equation of the parabola.
Q: What is the difference between a quadratic function and a linear function?
A: A quadratic function is a polynomial function of degree 2, while a linear function is a polynomial function of degree 1. Quadratic functions have a parabolic shape, while linear functions have a straight line shape.
Q: Can you provide some examples of quadratic functions with different axes of symmetry?
A: Here are some examples of quadratic functions with different axes of symmetry:
-
f(x) = x^2 + 4x + 6$ has an axis of symmetry of $x = -2$.
-
f(x) = x^2 - 2x + 6$ has an axis of symmetry of $x = 1$.
-
f(x) = x^2 + 6x + 6$ has an axis of symmetry of $x = -3$.
Q: How do I graph a quadratic function?
A: To graph a quadratic function, you can use the following steps:
- Find the axis of symmetry of the quadratic function.
- Find the vertex of the parabola.
- Plot the vertex on the graph.
- Plot two points on either side of the vertex.
- Draw a smooth curve through the points to form the parabola.
Q: What are some real-world applications of quadratic functions?
A: Quadratic functions have numerous real-world applications in various fields, including physics, engineering, and economics. For example, in physics, quadratic functions are used to describe the motion of objects under the influence of gravity. In engineering, quadratic functions are used to design structures that are resistant to stress and strain. In economics, quadratic functions are used to model the behavior of economic systems and make predictions about future trends.
Conclusion
In conclusion, the graph of a quadratic function is a powerful tool for modeling and analyzing real-world phenomena. By understanding the axis of symmetry and the vertex of a parabola, we can better analyze and model the behavior of quadratic functions. We hope that this Q&A guide has provided you with a better understanding of the graph of a quadratic function and its axis of symmetry.
Final Answer
The final answer is: