Vertex Form Overview
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Vertex form is a way to represent a quadratic function in a specific format that provides useful information about the function’s vertex. In this form, the vertex of the parabola is given by the coordinates (h, k), where h and k are constants that can be easily identified from the equation. Vertex form is also known as the completed square form because it involves completing the square to rewrite a quadratic function in a specific way.
The general form of a quadratic function is y = ax^2 + bx + c, where a, b, and c are constants. By completing the square, this function can be rewritten in the vertex form as y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola. The constants a, h, and k can be easily identified from the equation, making it easier to understand and analyze the function.
To convert a quadratic function from standard form to vertex form, we follow these steps:
Step 1: Factor out the coefficient of x^2
y = a(x^2 + (b/a)x) + c
Step 2: Complete the square by adding and subtracting (b/2a)^2 inside the parentheses
y = a(x^2 + (b/a)x + (b/(2a))^2 – (b/(2a))^2) + c
Step 3: Simplify the expression inside the parentheses
y = a((x + b/(2a))^2 – (b/(2a))^2) + c
Step 4: Expand and simplify
y = a(x + b/(2a))^2 – a(b/(2a))^2 + c
y = a(x + b/(2a))^2 – a(b^2/(4a^2)) + c
Step 5: Simplify further
y = a(x + b/(2a))^2 – (b^2/(4a)) + c
Now, the quadratic function is in vertex form, y = a(x-h)^2 + k, where h = -b/(2a) and k = c – (b^2/(4a)).
Vertex form provides valuable information about the vertex of the parabola and allows us to easily identify key characteristics of the function. The vertex of the parabola is given by the coordinates (h, k), where h = -b/(2a) and k = c – (b^2/(4a)). This means that the vertex form allows us to quickly determine the vertex without having to graph the function.
In addition to the vertex, the vertex form also shows the axis of symmetry of the parabola, which is given by the equation x = h. The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. Knowing the equation of the axis of symmetry can help us in graphing the function and analyzing its behavior.
Another advantage of the vertex form is that it clearly shows the direction and scale of the parabola. The coefficient a in front of the squared term determines the direction of the parabola: if a > 0, the parabola opens upwards, and if a 0, the vertex represents the minimum value of the function, and if a < 0, the vertex represents the maximum value of the function. By looking at the sign of the coefficient a, we can quickly determine whether the function has a minimum or maximum value.
In summary, vertex form is a useful way to represent a quadratic function that provides valuable information about the function’s vertex, axis of symmetry, direction, scale, and maximum/minimum value. By completing the square, we can easily convert a quadratic function from standard form to vertex form and gain insights into its properties. Vertex form simplifies the analysis and graphing of quadratic functions, making it a powerful tool in algebra and calculus.
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