A Parabola Intersects The $x$-axis At $x = 3$ And $ X = 9 X = 9 X = 9 [/tex]. What Is The $x$-coordinate Of The Parabola's Vertex? □ \square □
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Introduction
A parabola is a fundamental concept in mathematics, particularly in algebra and geometry. It is a quadratic equation that represents a U-shaped curve. In this article, we will explore the properties of a parabola that intersects the x-axis at two points, and we will find the x-coordinate of its vertex.
Understanding the Problem
The problem states that a parabola intersects the x-axis at two points, x = 3 and x = 9. This means that the parabola passes through these two points, and we can use this information to find the equation of the parabola.
The General Form of a Parabola
The general form of a parabola is given by the equation:
y = ax^2 + bx + c
where a, b, and c are constants. Since the parabola intersects the x-axis at x = 3 and x = 9, we know that the y-coordinate of these points is 0. Therefore, we can substitute these values into the equation to get:
0 = a(3)^2 + b(3) + c 0 = a(9)^2 + b(9) + c
Simplifying the Equations
Simplifying the equations, we get:
0 = 9a + 3b + c 0 = 81a + 9b + c
Subtracting the Equations
Subtracting the first equation from the second equation, we get:
0 = 72a + 6b
Dividing by 6
Dividing both sides of the equation by 6, we get:
0 = 12a + b
The Equation of the Parabola
Since the parabola intersects the x-axis at x = 3 and x = 9, we know that the equation of the parabola is of the form:
y = a(x - 3)(x - 9)
Expanding the Equation
Expanding the equation, we get:
y = a(x^2 - 12x + 27)
Comparing with the General Form
Comparing this equation with the general form of a parabola, we get:
ax^2 + bx + c = a(x^2 - 12x + 27)
Equating Coefficients
Equating the coefficients of the x^2 term, we get:
a = a
Equating the coefficients of the x term, we get:
b = -12a
Equating the constant term, we get:
c = 27a
The Vertex of the Parabola
The vertex of a parabola is given by the equation:
x = -b / 2a
Substituting the values of a and b, we get:
x = -(-12a) / (2a) x = 6
Conclusion
In this article, we have found the x-coordinate of the vertex of a parabola that intersects the x-axis at x = 3 and x = 9. The x-coordinate of the vertex is 6.
Final Answer
The final answer is 6.
Related Topics
- Quadratic Equations: Quadratic are a type of polynomial equation of degree two, which means the highest power of the variable is two. They have the general form ax^2 + bx + c = 0, where a, b, and c are constants.
- Parabolas: A parabola is a quadratic equation that represents a U-shaped curve. It has a vertex, which is the point where the parabola changes direction.
- Vertex Form: The vertex form of a parabola is given by the equation y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
References
- Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships. It involves the use of symbols, equations, and formulas to solve problems.
- Geometry: Geometry is a branch of mathematics that deals with the study of shapes, sizes, and positions of objects. It involves the use of points, lines, angles, and planes to describe and analyze geometric figures.
- Mathematics: Mathematics is a broad field that encompasses various branches, including algebra, geometry, calculus, and statistics. It involves the use of logical reasoning, mathematical operations, and problem-solving techniques to solve problems and make predictions.
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Introduction
In our previous article, we explored the properties of a parabola that intersects the x-axis at two points, and we found the x-coordinate of its vertex. In this article, we will answer some frequently asked questions related to parabolas and their properties.
Q&A
Q: What is a parabola?
A: A parabola is a quadratic equation that represents a U-shaped curve. It has a vertex, which is the point where the parabola changes direction.
Q: What is the general form of a parabola?
A: The general form of a parabola is given by the equation:
y = ax^2 + bx + c
where a, b, and c are constants.
Q: How do I find the equation of a parabola that intersects the x-axis at two points?
A: To find the equation of a parabola that intersects the x-axis at two points, you can use the following steps:
- Substitute the x-coordinates of the points into the general form of the parabola.
- Simplify the equations and solve for the constants a, b, and c.
- Use the values of a, b, and c to write the equation of the parabola.
Q: How do I find the x-coordinate of the vertex of a parabola?
A: To find the x-coordinate of the vertex of a parabola, you can use the following formula:
x = -b / 2a
where a and b are the coefficients of the x^2 and x terms, respectively.
Q: What is the vertex form of a parabola?
A: The vertex form of a parabola is given by the equation:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
Q: How do I convert the general form of a parabola to the vertex form?
A: To convert the general form of a parabola to the vertex form, you can complete the square by adding and subtracting the square of half the coefficient of the x term.
Q: What is the significance of the vertex of a parabola?
A: The vertex of a parabola is the point where the parabola changes direction. It is also the minimum or maximum point of the parabola, depending on the sign of the coefficient a.
Q: How do I find the y-coordinate of the vertex of a parabola?
A: To find the y-coordinate of the vertex of a parabola, you can substitute the x-coordinate of the vertex into the equation of the parabola.
Q: What is the relationship between the x-intercepts and the vertex of a parabola?
A: The x-intercepts of a parabola are the points where the parabola intersects the x-axis. The vertex of a parabola is the point where the parabola changes direction. The x-intercepts and the vertex are related by the equation:
x-intercept = -b / 2a
Q: do I use the x-intercepts to find the equation of a parabola?
A: To use the x-intercepts to find the equation of a parabola, you can substitute the x-coordinates of the x-intercepts into the general form of the parabola and solve for the constants a, b, and c.
Conclusion
In this article, we have answered some frequently asked questions related to parabolas and their properties. We have discussed the general form of a parabola, the vertex form of a parabola, and how to find the equation of a parabola that intersects the x-axis at two points.
Final Answer
The final answer is that parabolas are a fundamental concept in mathematics, and understanding their properties is essential for solving problems in algebra and geometry.
Related Topics
- Quadratic Equations: Quadratic are a type of polynomial equation of degree two, which means the highest power of the variable is two. They have the general form ax^2 + bx + c = 0, where a, b, and c are constants.
- Parabolas: A parabola is a quadratic equation that represents a U-shaped curve. It has a vertex, which is the point where the parabola changes direction.
- Vertex Form: The vertex form of a parabola is given by the equation y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
References
- Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships. It involves the use of symbols, equations, and formulas to solve problems.
- Geometry: Geometry is a branch of mathematics that deals with the study of shapes, sizes, and positions of objects. It involves the use of points, lines, angles, and planes to describe and analyze geometric figures.
- Mathematics: Mathematics is a broad field that encompasses various branches, including algebra, geometry, calculus, and statistics. It involves the use of logical reasoning, mathematical operations, and problem-solving techniques to solve problems and make predictions.