Understanding the Quadratic Parent Function: Equation, Graph, and Properties

Equation for Quadratic Parent Function

The equation for the general form of a quadratic parent function is:

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

In this equation, “a” represents the coefficient of the x^2 term, “b” represents the coefficient of the x term, and “c” represents the constant term

The equation for the general form of a quadratic parent function is:

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

In this equation, “a” represents the coefficient of the x^2 term, “b” represents the coefficient of the x term, and “c” represents the constant term.

The quadratic parent function is a basic form of the quadratic function, which represents a parabolic curve. It acts as a starting point from which different variations of quadratic functions can be derived by applying transformations such as shifting, stretching, or reflecting the graph.

Typically, the quadratic parent function is graphed as a symmetric U-shaped curve, called a parabola, symmetric about a vertical line known as the axis of symmetry.

The vertex of the parabola is an important point on the graph. It represents the minimum or maximum point of the function, depending on the value of “a”. If “a” is positive, the parabola opens upward, and the vertex represents the minimum point. If “a” is negative, the parabola opens downward, and the vertex represents the maximum point.

By manipulating the values of “a”, “b”, and “c” in the quadratic parent function equation, you can create different quadratic functions with different shapes and positions of the vertex.

To better understand this, let’s look at a few examples:

Example 1: f(x) = x^2
This is the basic quadratic parent function. It represents a simple U-shaped curve that opens upward. The vertex of this parabola is at the origin (0, 0).

Example 2: f(x) = 2x^2
In this case, “a” is 2, which means the parabola opens upward and is narrower than the basic quadratic parent function. The vertex of this parabola is still at the origin (0, 0).

Example 3: f(x) = -2x^2 + 3x – 1
Here, “a” is -2, so the parabola opens downward. The coefficients of “b” and “c” affect the position of the vertex. To find the vertex, you can use the formula x = -b/2a. Plugging in the values, we get x = -3/(-4) = 3/4. Substituting this value of x into the equation, we find that the y-coordinate of the vertex is f(3/4) = -2(3/4)^2 + 3(3/4) – 1 = -9/8 + 9/4 – 1 = -9/8 + 18/8 – 8/8 = 1/8. Therefore, the vertex of this parabola is at (3/4, 1/8).

It is important to note that the quadratic parent function serves as a reference or starting point for studying and graphing quadratic functions. By understanding its equation and properties, you can easily transform it to analyze and graph more complex quadratic functions.

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