Understanding the Standard Form of a Quadratic Function for Graphical Analysis and Problem Solving

Standard form of a quadratic function

The standard form of a quadratic function is written as:

f(x) = ax^2 + bx + c

In this form, “a”, “b”, and “c” are constants, with “a” representing the coefficient of the x^2 term, “b” representing the coefficient of the x term, and “c” representing the constant term

The standard form of a quadratic function is written as:

f(x) = ax^2 + bx + c

In this form, “a”, “b”, and “c” are constants, with “a” representing the coefficient of the x^2 term, “b” representing the coefficient of the x term, and “c” representing the constant term.

The standard form allows us to easily identify important characteristics of the quadratic function, such as the vertex, the axis of symmetry, and the y-intercept.

For example, if we have the quadratic function f(x) = 2x^2 + 3x – 4, we can determine the following:

1. Coefficient of the x^2 term (a) = 2
2. Coefficient of the x term (b) = 3
3. Constant term (c) = -4

Using this information, we can find:
– The vertex: The x-coordinate of the vertex, denoted by “h”, can be found using the formula h = -b/(2a). In this case, h = -3/(2*2) = -3/4. To find the y-coordinate, substitute this value into the equation: f(-3/4) = 2(-3/4)^2 + 3(-3/4) – 4 = -19/8. So, the vertex is (-3/4, -19/8).
– The axis of symmetry: It is a vertical line passing through the vertex. In this case, it is x = -3/4.
– The y-intercept: It is the point where the graph of the quadratic function intersects the y-axis. This occurs when x = 0. Substituting this into the equation gives: f(0) = 2(0)^2 +3(0) -4 = -4. So, the y-intercept is (0, -4).

Knowing the standard form of a quadratic function helps us understand its graphical behavior and allows us to easily analyze and solve problems involving quadratic equations.

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