Functions With Vertex On Y-Axis Analysis
- Introduction: Understanding Quadratic Functions and Vertices
- Key Concepts: Axis of Symmetry and Vertex
- Analyzing Function 1: f(x) = (x - 2)(x + 2)
- Analyzing Function 2: f(x) = x(x + 2)
- Analyzing Function 3: f(x) = (x + 1)(x - 2)
- Analyzing Function 4: f(x) = (x - 2)²
- Comparative Analysis: Vertex Positions
- Conclusion: Identifying Functions with Vertices on the y-axis
1. Introduction: Understanding Quadratic Functions and Vertices
In this comprehensive exploration, we aim to identify which quadratic function among the given options has its vertex situated on the y-axis. To achieve this, we will meticulously analyze each function, transforming them into standard quadratic form, and subsequently determining their respective vertices. The functions under consideration are:
f(x) = (x - 2)(x + 2)
f(x) = x(x + 2)
f(x) = (x + 1)(x - 2)
f(x) = (x - 2)²
Understanding quadratic functions is pivotal in mathematics, especially in algebra and calculus. A quadratic function is generally expressed in the form f(x) = ax² + bx + c
, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graphical representation of a quadratic function is a parabola, a U-shaped curve that opens upwards if a > 0
and downwards if a < 0
. The vertex is a crucial point on the parabola; it represents the minimum value of the function if the parabola opens upwards, and the maximum value if it opens downwards.
The vertex of a parabola is the point where the parabola changes direction. This point is significant because it provides critical information about the function's behavior. The vertex form of a quadratic function, f(x) = a(x - h)² + k
, directly reveals the vertex coordinates as (h, k)
. In this form, h
represents the x-coordinate of the vertex, and k
represents the y-coordinate. Converting a quadratic function into vertex form simplifies the process of finding the vertex. Alternatively, the vertex can be found using the standard form f(x) = ax² + bx + c
. The x-coordinate of the vertex, h
, can be calculated using the formula h = -b / (2a)
, and the y-coordinate, k
, can be found by substituting h
back into the function, k = f(h)
.
The y-axis is defined by the equation x = 0
. Therefore, for a vertex to lie on the y-axis, its x-coordinate must be zero. This condition is crucial in determining which of the given functions meets the requirement. By systematically analyzing each function and calculating its vertex, we can identify the function(s) that satisfy this condition. This involves expanding each function, converting it into standard form, and then applying the vertex formula or completing the square to find the vertex coordinates. This detailed process will ensure a thorough understanding of how the vertex is determined and its significance in the context of quadratic functions.
2. Key Concepts: Axis of Symmetry and Vertex
Before diving into the analysis of the specific functions, it is essential to solidify our understanding of key concepts such as the axis of symmetry and the vertex of a parabola. These concepts are fundamental to identifying which function has a vertex on the y-axis. The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. This line is crucial because it provides a reference point for the parabola's symmetry and helps in locating the vertex.
The axis of symmetry is defined by the equation x = h
, where h
is the x-coordinate of the vertex. This means that the x-coordinate of the vertex is the same as the value of x
in the equation of the axis of symmetry. The axis of symmetry is always equidistant from the roots (or zeros) of the quadratic function, if they exist. In other words, if a quadratic function has two distinct real roots, the axis of symmetry will lie exactly in the middle of these roots. This property is particularly useful when the function is given in factored form, as the roots can be easily identified.
The vertex of a parabola, as mentioned earlier, is the point where the parabola changes direction. It is the minimum point if the parabola opens upwards (i.e., a > 0
) and the maximum point if the parabola opens downwards (i.e., a < 0
). The vertex lies on the axis of symmetry, and its coordinates (h, k)
are essential for understanding the behavior of the quadratic function. The x-coordinate h
determines the horizontal position of the vertex, while the y-coordinate k
represents the minimum or maximum value of the function. The vertex form of a quadratic function, f(x) = a(x - h)² + k
, directly provides the vertex coordinates, making it a convenient form for analysis.
To find the vertex using the standard form f(x) = ax² + bx + c
, we first determine the x-coordinate h
using the formula h = -b / (2a)
. Then, we substitute this value back into the function to find the y-coordinate k = f(h)
. The vertex (h, k)
is a critical point for graphing the parabola and understanding its characteristics. Knowing the vertex and the direction in which the parabola opens (upwards or downwards) allows us to sketch the graph accurately and predict the function's behavior over different intervals.
Understanding the relationship between the axis of symmetry and the vertex is crucial for solving problems related to quadratic functions. For a vertex to lie on the y-axis, its x-coordinate must be zero. This means that the axis of symmetry must be the y-axis itself (i.e., x = 0
). In the context of the given problem, we need to identify which quadratic function has an axis of symmetry that coincides with the y-axis. This involves calculating the x-coordinate of the vertex for each function and determining whether it is equal to zero. If the x-coordinate of the vertex is zero, then the vertex lies on the y-axis, and that function is the solution.
3. Analyzing Function 1: f(x) = (x - 2)(x + 2)
Our first task is to meticulously analyze the quadratic function f(x) = (x - 2)(x + 2) to determine if its vertex lies on the y-axis. This involves several steps, including expanding the function, identifying its standard form, and then calculating the vertex coordinates. By systematically following these steps, we can accurately determine the position of the vertex and whether it meets the condition of lying on the y-axis.
Expanding the Function
The initial step in analyzing f(x) = (x - 2)(x + 2)
is to expand the function. This involves applying the distributive property (also known as the FOIL method) to multiply the two binomials. The function is in factored form, which is useful for identifying the roots of the equation but not for directly determining the vertex. Expanding the function allows us to rewrite it in the standard quadratic form f(x) = ax² + bx + c
, which is necessary for calculating the vertex using the standard formula.
Expanding (x - 2)(x + 2)
involves multiplying each term in the first binomial by each term in the second binomial:
x * x = x²
x * 2 = 2x
-2 * x = -2x
-2 * 2 = -4
Combining these terms, we get:
f(x) = x² + 2x - 2x - 4
Simplifying the expression by combining like terms (2x
and -2x
cancel each other out), we obtain the standard quadratic form:
f(x) = x² - 4
This expanded form is much easier to analyze for its coefficients and to apply the vertex formula. The standard form f(x) = x² - 4
clearly shows that a = 1
, b = 0
, and c = -4
. These coefficients are crucial for the next step, which is finding the vertex of the parabola.
Finding the Vertex
With the function now in the standard form f(x) = x² - 4
, we can proceed to find the vertex. The vertex of a parabola is the point at which it changes direction, and its coordinates (h, k)
can be determined using specific formulas. The x-coordinate h
of the vertex is given by the formula h = -b / (2a)
, where a
and b
are the coefficients from the standard quadratic form f(x) = ax² + bx + c
. In this case, a = 1
and b = 0
.
Plugging these values into the formula, we get:
h = -0 / (2 * 1) = 0
So, the x-coordinate of the vertex is 0
. This means that the axis of symmetry for this parabola is the y-axis (x = 0
).
To find the y-coordinate k
of the vertex, we substitute the value of h
back into the function f(x)
:
k = f(0) = (0)² - 4 = -4
Therefore, the vertex of the function f(x) = x² - 4
is (0, -4)
. This point is critical for understanding the parabola's position and orientation in the coordinate plane. The next step is to determine whether this vertex lies on the y-axis, which is the primary goal of our analysis.
Determining the Vertex Location
Now that we have calculated the vertex of the function f(x) = x² - 4 to be (0, -4)
, we can determine its location in relation to the y-axis. The key criterion for a point to lie on the y-axis is that its x-coordinate must be zero. In this case, the x-coordinate of the vertex is indeed 0
.
Since the x-coordinate of the vertex is 0
, the vertex (0, -4)
lies on the y-axis. This confirms that the function f(x) = (x - 2)(x + 2)
has a vertex on the y-axis. The y-coordinate -4
tells us that the vertex is located 4 units below the x-axis. The parabola opens upwards because the coefficient of the x²
term (a
) is positive (a = 1
).
In summary, the vertex of the function f(x) = (x - 2)(x + 2)
is (0, -4)
, which lies on the y-axis. This makes f(x) = (x - 2)(x + 2)
one of the functions that meet the condition specified in the problem. To ensure we have identified all such functions, we must now analyze the remaining functions in a similar manner. This involves expanding the functions, finding their vertices, and checking if the x-coordinate of the vertex is zero. The comprehensive analysis of each function will allow us to draw a definitive conclusion about which functions have vertices on the y-axis.
4. Analyzing Function 2: f(x) = x(x + 2)
Next, we turn our attention to the quadratic function f(x) = x(x + 2). Our objective remains consistent: to determine whether the vertex of this function lies on the y-axis. This process will mirror the analysis conducted for the first function, involving expansion, identification of standard form, and calculation of vertex coordinates. Each step is crucial in accurately locating the vertex and determining its position relative to the y-axis.
Expanding the Function
As with the previous function, the initial step in analyzing f(x) = x(x + 2)
is to expand the function. This is necessary to convert the function into the standard quadratic form f(x) = ax² + bx + c
, which is essential for applying the vertex formula. Expanding the function involves distributing x
across the terms inside the parenthesis:
f(x) = x * x + x * 2
Performing the multiplication, we get:
f(x) = x² + 2x
This expanded form is now in the standard quadratic form, where we can easily identify the coefficients. In this case, a = 1
, b = 2
, and c = 0
. These coefficients are vital for calculating the vertex coordinates in the next step. Having the function in standard form simplifies the process of finding the vertex, which is a key element in determining whether it lies on the y-axis.
Finding the Vertex
Now that we have the function in the standard form f(x) = x² + 2x
, we can proceed to find its vertex. The x-coordinate h
of the vertex is given by the formula h = -b / (2a)
. In this function, a = 1
and b = 2
. Plugging these values into the formula, we get:
h = -2 / (2 * 1) = -1
So, the x-coordinate of the vertex is -1
. This indicates that the axis of symmetry for this parabola is the vertical line x = -1
, which is not the y-axis.
To find the y-coordinate k
of the vertex, we substitute the value of h
back into the function f(x)
:
k = f(-1) = (-1)² + 2(-1) = 1 - 2 = -1
Therefore, the vertex of the function f(x) = x² + 2x
is (-1, -1)
. This point is crucial for understanding the parabola's position in the coordinate plane. The next step is to determine whether this vertex lies on the y-axis, which is our primary objective.
Determining the Vertex Location
Having calculated the vertex of the function f(x) = x² + 2x to be (-1, -1)
, we now need to determine its location relative to the y-axis. The condition for a point to lie on the y-axis is that its x-coordinate must be zero. In this case, the x-coordinate of the vertex is -1
.
Since the x-coordinate of the vertex is -1
, which is not equal to zero, the vertex (-1, -1)
does not lie on the y-axis. This means that the function f(x) = x(x + 2)
does not meet the condition specified in the problem. The vertex is located 1 unit to the left of the y-axis and 1 unit below the x-axis. The parabola opens upwards because the coefficient of the x²
term (a
) is positive (a = 1
).
In summary, the vertex of the function f(x) = x(x + 2)
is (-1, -1)
, which does not lie on the y-axis. Therefore, this function is not a solution to the problem. We must continue our analysis with the remaining functions to identify all those that have vertices on the y-axis. This methodical approach ensures that we accurately determine which functions meet the given condition.
5. Analyzing Function 3: f(x) = (x + 1)(x - 2)
Our analysis now shifts to the quadratic function f(x) = (x + 1)(x - 2). As with the previous functions, the goal remains to determine if the vertex of this function lies on the y-axis. This involves the same systematic process of expanding the function, converting it to standard form, and calculating the vertex coordinates. Each step is vital for accurately determining the position of the vertex and whether it satisfies the given condition.
Expanding the Function
The initial step in analyzing f(x) = (x + 1)(x - 2)
is to expand the function. This involves applying the distributive property (FOIL method) to multiply the two binomials. Expanding the function is essential to rewrite it in the standard quadratic form f(x) = ax² + bx + c
, which is necessary for calculating the vertex using the standard formula.
Expanding (x + 1)(x - 2)
involves multiplying each term in the first binomial by each term in the second binomial:
x * x = x²
x * -2 = -2x
1 * x = x
1 * -2 = -2
Combining these terms, we get:
f(x) = x² - 2x + x - 2
Simplifying the expression by combining like terms (-2x
and x
), we obtain the standard quadratic form:
f(x) = x² - x - 2
This expanded form clearly shows that a = 1
, b = -1
, and c = -2
. These coefficients are crucial for the next step, which is finding the vertex of the parabola. Having the function in standard form makes it straightforward to apply the vertex formula.
Finding the Vertex
With the function now in the standard form f(x) = x² - x - 2
, we can proceed to find the vertex. The x-coordinate h
of the vertex is given by the formula h = -b / (2a)
. In this function, a = 1
and b = -1
. Plugging these values into the formula, we get:
h = -(-1) / (2 * 1) = 1 / 2 = 0.5
So, the x-coordinate of the vertex is 0.5
. This means that the axis of symmetry for this parabola is the vertical line x = 0.5
, which is not the y-axis.
To find the y-coordinate k
of the vertex, we substitute the value of h
back into the function f(x)
:
k = f(0.5) = (0.5)² - (0.5) - 2 = 0.25 - 0.5 - 2 = -2.25
Therefore, the vertex of the function f(x) = x² - x - 2
is (0.5, -2.25)
. This point is essential for understanding the parabola's position in the coordinate plane. The next step is to determine whether this vertex lies on the y-axis, which is our primary goal.
Determining the Vertex Location
Having calculated the vertex of the function f(x) = x² - x - 2 to be (0.5, -2.25)
, we now need to determine its location relative to the y-axis. The condition for a point to lie on the y-axis is that its x-coordinate must be zero. In this case, the x-coordinate of the vertex is 0.5
.
Since the x-coordinate of the vertex is 0.5
, which is not equal to zero, the vertex (0.5, -2.25)
does not lie on the y-axis. This means that the function f(x) = (x + 1)(x - 2)
does not meet the condition specified in the problem. The vertex is located 0.5 units to the right of the y-axis and 2.25 units below the x-axis. The parabola opens upwards because the coefficient of the x²
term (a
) is positive (a = 1
).
In summary, the vertex of the function f(x) = (x + 1)(x - 2)
is (0.5, -2.25)
, which does not lie on the y-axis. Therefore, this function is not a solution to the problem. We must continue our analysis with the remaining function to identify any other functions that have vertices on the y-axis. This systematic approach ensures that we accurately determine which functions meet the given condition.
6. Analyzing Function 4: f(x) = (x - 2)²
Our final analysis focuses on the quadratic function f(x) = (x - 2)². As with the previous functions, our goal is to determine if the vertex of this function lies on the y-axis. We will follow the same methodical process of expanding the function, converting it to standard form, and calculating the vertex coordinates. This consistent approach ensures an accurate determination of the vertex's position and whether it meets the required condition.
Expanding the Function
The first step in analyzing f(x) = (x - 2)²
is to expand the function. This involves multiplying the binomial (x - 2)
by itself. Expanding the function is necessary to rewrite it in the standard quadratic form f(x) = ax² + bx + c
, which is essential for calculating the vertex using the standard formula. Alternatively, we can recognize this as a perfect square trinomial, which can simplify the expansion process.
Expanding (x - 2)²
means multiplying (x - 2)(x - 2)
:
x * x = x²
x * -2 = -2x
-2 * x = -2x
-2 * -2 = 4
Combining these terms, we get:
f(x) = x² - 2x - 2x + 4
Simplifying the expression by combining like terms (-2x
and -2x
), we obtain the standard quadratic form:
f(x) = x² - 4x + 4
This expanded form clearly shows that a = 1
, b = -4
, and c = 4
. These coefficients are crucial for the next step, which is finding the vertex of the parabola. Having the function in standard form allows us to easily apply the vertex formula.
Finding the Vertex
With the function now in the standard form f(x) = x² - 4x + 4
, we can proceed to find the vertex. The x-coordinate h
of the vertex is given by the formula h = -b / (2a)
. In this function, a = 1
and b = -4
. Plugging these values into the formula, we get:
h = -(-4) / (2 * 1) = 4 / 2 = 2
So, the x-coordinate of the vertex is 2
. This means that the axis of symmetry for this parabola is the vertical line x = 2
, which is not the y-axis.
To find the y-coordinate k
of the vertex, we substitute the value of h
back into the function f(x)
:
k = f(2) = (2)² - 4(2) + 4 = 4 - 8 + 4 = 0
Therefore, the vertex of the function f(x) = x² - 4x + 4
is (2, 0)
. This point is essential for understanding the parabola's position in the coordinate plane. The next step is to determine whether this vertex lies on the y-axis, which is our primary objective.
Determining the Vertex Location
Having calculated the vertex of the function f(x) = x² - 4x + 4 to be (2, 0)
, we now need to determine its location relative to the y-axis. The condition for a point to lie on the y-axis is that its x-coordinate must be zero. In this case, the x-coordinate of the vertex is 2
.
Since the x-coordinate of the vertex is 2
, which is not equal to zero, the vertex (2, 0)
does not lie on the y-axis. This means that the function f(x) = (x - 2)²
does not meet the condition specified in the problem. The vertex is located 2 units to the right of the y-axis and on the x-axis. The parabola opens upwards because the coefficient of the x²
term (a
) is positive (a = 1
).
In summary, the vertex of the function f(x) = (x - 2)²
is (2, 0)
, which does not lie on the y-axis. Therefore, this function is not a solution to the problem. With the analysis of all four functions now complete, we can draw a final conclusion about which function(s) have vertices on the y-axis.
7. Comparative Analysis: Vertex Positions
Having meticulously analyzed each of the four quadratic functions, a comparative analysis of their vertex positions is essential to consolidate our findings. This step allows us to clearly see which functions meet the criterion of having a vertex on the y-axis. We will summarize the vertex coordinates for each function and then draw a conclusion based on this information. This comparative analysis provides a clear and concise overview of our results.
-
Function 1: f(x) = (x - 2)(x + 2)
- Expanded Form:
f(x) = x² - 4
- Vertex:
(0, -4)
- Location: The vertex lies on the y-axis because its x-coordinate is 0.
- Expanded Form:
-
Function 2: f(x) = x(x + 2)
- Expanded Form:
f(x) = x² + 2x
- Vertex:
(-1, -1)
- Location: The vertex does not lie on the y-axis because its x-coordinate is -1.
- Expanded Form:
-
Function 3: f(x) = (x + 1)(x - 2)
- Expanded Form:
f(x) = x² - x - 2
- Vertex:
(0.5, -2.25)
- Location: The vertex does not lie on the y-axis because its x-coordinate is 0.5.
- Expanded Form:
-
Function 4: f(x) = (x - 2)²
- Expanded Form:
f(x) = x² - 4x + 4
- Vertex:
(2, 0)
- Location: The vertex does not lie on the y-axis because its x-coordinate is 2.
- Expanded Form:
From this comparative analysis, it is evident that only one function has a vertex that lies on the y-axis. This function is f(x) = (x - 2)(x + 2)
, which has a vertex at (0, -4)
. The other functions have vertices with non-zero x-coordinates, meaning they are located either to the left or right of the y-axis. This clear distinction allows us to draw a firm conclusion about which function satisfies the given condition.
8. Conclusion: Identifying Functions with Vertices on the y-axis
In conclusion, after a thorough and systematic analysis of the given quadratic functions, we have identified the function that has a vertex on the y-axis. Our method involved expanding each function to its standard form, calculating the vertex coordinates, and then determining whether the x-coordinate of the vertex is zero. This comprehensive approach ensures the accuracy and reliability of our results. Our conclusion definitively answers the question posed in the problem.
Based on our analysis, the function that has a vertex on the y-axis is:
f(x) = (x - 2)(x + 2)
This function, when expanded, simplifies to f(x) = x² - 4
. The vertex of this parabola is located at the point (0, -4)
, which lies directly on the y-axis. This result confirms that the axis of symmetry for this parabola is the y-axis itself, as the x-coordinate of the vertex is zero. The parabola opens upwards since the coefficient of the x²
term is positive (a = 1
).
The other functions analyzed do not have vertices on the y-axis:
f(x) = x(x + 2)
has a vertex at(-1, -1)
. This parabola's vertex is located to the left of the y-axis.f(x) = (x + 1)(x - 2)
has a vertex at(0.5, -2.25)
. This parabola's vertex is located to the right of the y-axis.f(x) = (x - 2)²
has a vertex at(2, 0)
. This parabola's vertex is also located to the right of the y-axis.
Therefore, we can definitively state that only the function f(x) = (x - 2)(x + 2)
has its vertex on the y-axis. This conclusion is supported by our step-by-step analysis and comparative review of all the given functions. Our analysis has provided a clear and comprehensive answer to the problem, demonstrating the importance of understanding quadratic functions and their properties.