Problem 7
Question
Identify the vertex of each parabola. $$ f(x)=\frac{1}{3} x^{2} $$
Step-by-Step Solution
Verified Answer
The vertex is at (0, 0).
1Step 1: Understand the form of the quadratic equation
The given function is in the form of a quadratic equation, which is generally written as \( f(x) = ax^2 + bx + c \).In this problem, \( f(x) = \frac{1}{3} x^{2} \) with \( a = \frac{1}{3} \), \( b = 0 \), and \( c = 0 \).
2Step 2: Identify the vertex formula
The vertex of a parabola given by \( f(x) = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex: \( x = -\frac{b}{2a} \).
3Step 3: Calculate the x-coordinate of the vertex
Substitute the values of \( a \) and \( b \) into the vertex formula: \( x = -\frac{0}{2 \times \frac{1}{3}} = 0 \). Therefore, the x-coordinate of the vertex is \( x = 0 \).
4Step 4: Find the y-coordinate of the vertex
Substitute \( x = 0 \) back into the original function to find the y-coordinate: \( f(0) = \frac{1}{3} (0)^2 = 0 \). Therefore, the y-coordinate of the vertex is \( y = 0 \).
5Step 5: Combine the coordinates
The vertex of the parabola is located at the point \( (0, 0) \).
Key Concepts
Quadratic EquationVertex FormulaParabolic Function
Quadratic Equation
A quadratic equation is a type of polynomial equation in the form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. It graphs as a parabola, which is a U-shaped curve.
In our exercise, the quadratic equation is \( f(x) = \frac{1}{3} x^2 \). Here, we identify \( a = \frac{1}{3} \), \( b = 0 \), and \( c = 0 \). This particular equation is a simple form known as a 'monic' quadratic, because it does not have \( bx \) or \( c \) terms.
Quadratic equations have key features such as their vertex (the highest or lowest point), axis of symmetry, and they open upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)). Understanding these features is essential for solving quadratic problems.
In our exercise, the quadratic equation is \( f(x) = \frac{1}{3} x^2 \). Here, we identify \( a = \frac{1}{3} \), \( b = 0 \), and \( c = 0 \). This particular equation is a simple form known as a 'monic' quadratic, because it does not have \( bx \) or \( c \) terms.
Quadratic equations have key features such as their vertex (the highest or lowest point), axis of symmetry, and they open upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)). Understanding these features is essential for solving quadratic problems.
Vertex Formula
The vertex of a parabola is a critical point where the curve changes direction. For parabolas that open upwards, it is the minimum point; for parabolas that open downwards, it is the maximum point. To find the vertex of a quadratic function \( f(x) = ax^2 + bx + c \), we use the vertex formula for the x-coordinate: \( x = -\frac{b}{2a} \).
Once we have the x-coordinate, we substitute it back into the original function to find the y-coordinate, thus pinpointing the exact vertex. In our exercise, the function \( f(x) = \frac{1}{3} x^2 \) has \( b = 0 \), leading to an x-coordinate of \( x = -\frac{0}{2 \times \frac{1}{3}} = 0 \). Substituting \( x = 0 \) back into \( f(x) \), we get \( y = \frac{1}{3} (0)^2 = 0 \). Therefore, the vertex is \( (0, 0) \).
Understanding how to use the vertex formula is crucial when dealing with quadratic functions, as it helps in identifying the most significant point on the graph.
Once we have the x-coordinate, we substitute it back into the original function to find the y-coordinate, thus pinpointing the exact vertex. In our exercise, the function \( f(x) = \frac{1}{3} x^2 \) has \( b = 0 \), leading to an x-coordinate of \( x = -\frac{0}{2 \times \frac{1}{3}} = 0 \). Substituting \( x = 0 \) back into \( f(x) \), we get \( y = \frac{1}{3} (0)^2 = 0 \). Therefore, the vertex is \( (0, 0) \).
Understanding how to use the vertex formula is crucial when dealing with quadratic functions, as it helps in identifying the most significant point on the graph.
Parabolic Function
A parabolic function is a specific type of quadratic function that results in a graph that is shaped like a parabola. These functions are the foundation for many concepts in algebra and calculus.
In our example, \( f(x) = \frac{1}{3} x^2 \) is a parabolic function. Since the coefficient \( a = \frac{1}{3} \) is positive, the parabola opens upwards. The vertex of this parabola is at the point \( (0,0) \), as we've calculated earlier. This point is the minimum point on the graph because the parabola opens upwards.
The graph of a parabolic function is symmetric about a line called the 'axis of symmetry,' which, for our function, is the vertical line \( x = 0 \). This symmetry property is helpful when graphing the function and understanding its behavior.
Parabolic functions are used in various fields such as physics for projectile motion, engineering for bridge arches, and even finance for modeling certain economic behaviors.
In our example, \( f(x) = \frac{1}{3} x^2 \) is a parabolic function. Since the coefficient \( a = \frac{1}{3} \) is positive, the parabola opens upwards. The vertex of this parabola is at the point \( (0,0) \), as we've calculated earlier. This point is the minimum point on the graph because the parabola opens upwards.
The graph of a parabolic function is symmetric about a line called the 'axis of symmetry,' which, for our function, is the vertical line \( x = 0 \). This symmetry property is helpful when graphing the function and understanding its behavior.
Parabolic functions are used in various fields such as physics for projectile motion, engineering for bridge arches, and even finance for modeling certain economic behaviors.
Other exercises in this chapter
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