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Vertex Formula
The vertex formula is a mathematical expression used to find the coordinates of the vertex of a parabola in the form of a quadratic function. A quadratic function is typically written in the form:
where ‘a’, ‘b’, and ‘c’ are constants, and ‘x’ is the variable.
The vertex of a parabola is the point where it reaches either a minimum or a maximum value, depending on the coefficient ‘a’. If ‘a’ is positive, the parabola opens upwards and has a minimum value at the vertex. If ‘a’ is negative, the parabola opens downward and has a maximum value at the vertex.
The formula to find the x-coordinate of the vertex is given by:
x_vertice = -b / (2a)
To find the y-coordinate of the vertex, you substitute the x-coordinate (x_vertex) back into the original equation:
y_vertice = f(x_vertice) = f(-b / (2a)) = a * (-b / (2a))^2 + b * (-b / (2a)) + c
Simplifying this expression will give you the y-coordinate of the vertex.
It is worth noting that the vertex formula is especially useful when dealing with parabolas, and it helps to quickly find the vertex without graphing the function. The vertex represents the “turning point” of the parabola, and it lies on the axis of symmetry of the parabola, which is the vertical line passing through the vertex and dividing the parabola into two symmetrical halves.
What is Vertex Formula?
The term “Vertex Formula” typically refers to the formula used to find the vertex of a parabola, which is the point where the parabola reaches its highest or lowest point, depending on its orientation.
A parabola is a U-shaped curve that can open up or down and is defined by a quadratic equation of the form:
Where:
y is the output or dependent variable (usually representing the vertical axis).
x is the input or independent variable (usually representing the horizontal axis).
a, b, and c are constants, with ‘a’ not equal to 0.
The vertex formula is used to find the x-coordinate (h) and the y-coordinate (k) of the vertex, represented as the point (h, k). The vertex of a parabola can be calculated using the following formula:
- h = -b / (2a)
- k = f(h) = a(h)^2 + b(h) + c
Where:
‘h’ is the x-coordinate of the vertex.
‘k’ is the y-coordinate of the vertex.
‘a’, ‘b’, and ‘c’ are the coefficients of the quadratic equation.
To summarize, the vertex formula helps you find the vertex of a parabola without graphing it. By identifying the vertex, you can determine the axis of symmetry and better understand the behavior of the parabola.
How to Use Vertex Formula?
The vertex formula, also known as the vertex form, is used to express a quadratic function in a simplified form that easily reveals the vertex of the parabola. A quadratic function is a polynomial of degree 2 and has the general form:
where:
f(x) is the output or dependent variable (usually represented on the y-axis).
x is the input or independent variable (usually represented on the x-axis).
a, b, and c are constants, where ‘a’ is the coefficient of the x^2 term.
The vertex formula takes the form:
where:
(h, k) represents the coordinates of the vertex of the parabola.
To use the vertex formula, follow these steps:
Step 1: Identify the values of ‘a’, ‘h’ and ‘k’ of the given quadratic function.
Step 2: Use the values of ‘h’ and ‘k’ to determine the coordinates of the vertex. The vertex is the point (h, k).
Here is a step by step example:
Example: Given the quadratic function f(x) = 2(x – 3)^2 + 5, find the coordinates of the vertex.
Step 1: Identify the values of ‘a’, ‘h’ and ‘k’.
‘a’ is the coefficient of the x^2 term, so a = 2.
‘h’ is the x-coordinate of the vertex, so h = 3.
‘k’ is the y-coordinate of the vertex, so k = 5.
Step 2: Find the vertex coordinates (h, k).
In this example, the vertex has coordinates (h, k) = (3, 5).
So, the vertex of the parabola represented by the given function is at the point (3, 5).
The vertex form of a quadratic function is especially useful when graphing parabolas or analyzing their properties, because it allows you to quickly identify the vertex and other important features of the parabola.
Derivation of Vertex Formulas
To derive the vertex formula for a quadratic function in the form y = ax^2 + bx + c, we complete the square. The vertex form of a quadratic function is given by y = a(x – h)^2 + k, where (h, k) represents the coordinates of the vertex.
Step 1: Start with the quadratic function in the standard form: y = ax^2 + bx + c.
Step 2: To complete the square, we add and subtract (b/2a)^2 inside the parentheses. This expression comes from halving the coefficient of x (b) and squaring it.
Step 3: Rewrite the quadratic expression with the added and subtracted terms:
y = ax^2 + bx + (b/2a)^2 – (b/2a)^2 + c
Step 4: Factor the first three terms inside the brackets:
y = a(x^2 + (b/a)x + (b/2a)^2) – (b/2a)^2 + c
Step 5: Now, notice that the expression inside the brackets can be written as a perfect square trinomial: (x + (b/2a))^2 = x^2 + (b/a)x + (b/2a)^2.
Step 6: Rewrite the equation with the perfect square trinomial:
y = a(x + (b/2a))^2 – (b/2a)^2 + c
Step 7: To put it in vertex form, factor ‘a’ of the perfect square trinomial:
y = a(x + (b/2a))^2 + (4ac – b^2)/(4a^2)
Step 8: Now, the vertex form of the quadratic function is y = a(x – h)^2 + k, where h = -b/2a and k = (4ac – b^2)/(4a^2).
So, the vertex of the parabola is at the point (h, k) = (-b/2a, (4ac – b^2)/(4a^2)).
How to Find the Vertex of a Function?
To find the vertex of a function, you must follow these steps:
Understand the vertex form of the function:
The vertex form of a quadratic function is given by:
where:
a is the coefficient of the quadratic term,
(h, k) represents the coordinates of the vertex.
Identify the coefficients:
Look at the given function and identify the values of a, h and k. The coefficient ‘a’ determines whether the parabola opens up (a > 0) or down (a
Find the x-coordinate of the vertex:
The x-coordinate of the vertex (h) is simply the value that makes the expression inside the brackets equal to zero. So, set (x – h) = 0 and solve for x.
Calculate the y-coordinate of the vertex:
Substitute the x-coordinate (h) back into the original function to find the corresponding y-coordinate (k).
Write the vertex:
Once you have the values of h and k, write the vertex as (h, k).
Remember that the vertex of a function is a point on the graph where the function reaches its maximum or minimum value, depending on the orientation of the parabola.
Example:
Let’s find the vertex of the quadratic function: f(x) = 2(x – 3)^2 + 5.
The function is already in vertex form: f(x) = 2(x – 3)^2 + 5.
Coefficients: a = 2, h = 3, k = 5.
Find the x-coordinate of the vertex:
Set (x – h) = 0:
x – 3 = 0
x = 3
Calculate the y-coordinate of the vertex:
Substitute x = 3 back into the function:
f(3) = 2(3 – 3)^2 + 5
f(3) = 2(0)^2 + 5
f(3) = 5
Write the vertex:
The vertex is (3, 5).
So, the vertex of the given function is at (3, 5).
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