The graph of the quadratic function g is obtained from the...

The graph of the quadratic function $g$ is obtained from the graph of $f(x)=x^{2}$ by shifting it horizontally 4 units to the left, then vertically stretching it by a factor of $3,$ and then shifting vertically 2 units upward. What is the rule of the function $g$ ? What is the vertex of its graph?

Key Concepts

Function Transformations

This concept involves operations that alter the position or shape of a function's graph. Transformations include translations (shifts), stretches, compressions, and reflections, which allow one to modify a basic function to achieve a desired graph or behavior without changing its fundamental properties.

Horizontal Translation

A horizontal translation shifts the graph of a function left or right. This is achieved by replacing the input variable x with (x - h), where h indicates the direction and magnitude of the shift. A positive h shifts the graph to the right, while a negative h shifts it to the left.

Vertical Stretch

Vertical stretching is a transformation that scales the output values of a function by a constant factor. Multiplying the function by a factor greater than 1 stretches the graph away from the horizontal axis, making it steeper, while a factor between 0 and 1 compresses it toward the axis.

Vertical Translation

Vertical translation involves shifting the entire graph of a function upward or downward by adding or subtracting a constant from the function's output. This simply moves the graph without altering its shape or orientation.

Vertex of a Quadratic Function

For quadratic functions, the vertex represents the highest or lowest point of the parabola. When a quadratic is expressed in vertex form, the coordinates of the vertex are clearly indicated. Understanding vertex movement due to transformations is essential to graphing and analyzing quadratic functions.

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