Mastering Quadratic Functions: Understanding the Quadratic Parent Function and its Transformations

Quadratic Parent Function

The quadratic parent function, also known as the parent equation, is a standard form quadratic equation with the form y = x^2

The quadratic parent function, also known as the parent equation, is a standard form quadratic equation with the form y = x^2. It serves as the basis for various quadratic functions and graphs.

In this equation, the variable x represents the input or independent variable, and the variable y represents the output or dependent variable. The exponent of 2 implies that the equation is of degree 2, meaning it represents a parabola when graphed.

The graph of the quadratic parent function is a U-shaped curve called a parabola. The vertex of the parabola is located at the origin (0, 0), which is the lowest or highest point of the curve, depending on the direction it opens. In this case, since the coefficient attached to x^2 is positive (1), the parabola opens upwards.

The quadratic parent function can undergo various transformations to produce different quadratic functions. Some common transformations include:

1. Vertical Translation: a constant term, h, is added or subtracted to shift the graph vertically. For example, the quadratic function y = x^2 + 2 would shift the graph upward by 2 units, while y = x^2 – 3 would shift the graph downward by 3 units.

2. Horizontal Translation: a constant term, h, is added or subtracted inside the function to shift the graph horizontally. For example, the quadratic function y = (x – 2)^2 would shift the graph 2 units to the right, while y = (x + 3)^2 would shift the graph 3 units to the left.

3. Vertical Stretch or Compression: a coefficient, a, is multiplied by the function to stretch or compress the graph vertically. If a > 1, the graph is stretched vertically; if 0 < a < 1, the graph is compressed vertically. For example, y = 3x^2 would stretch the graph vertically by a factor of 3, while y = (1/2)x^2 would compress the graph vertically by a factor of 1/2. 4. Horizontal Stretch or Compression: the reciprocal of a coefficient, a, is multiplied by the function to stretch or compress the graph horizontally. If a > 1, the graph is compressed horizontally; if 0 < a < 1, the graph is stretched horizontally. For example, y = (2x)^2 would compress the graph horizontally by a factor of 2, while y = (1/3x)^2 would stretch the graph horizontally by a factor of 3. By combining these transformations, you can create quadratic functions that have different shapes and positions compared to the quadratic parent function.

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