27-32 : A function f is given, and the indicated...

$27-32$ : A function $f$ is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph.
$f(x)=|x| ;$ shift to the left 1 unit, stretch vertically by a factor of $3,$ and shift upward 10 units

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Key Concepts

Absolute Value Function

The absolute value function, typically defined as f(x) = |x|, produces a V-shaped graph that reflects the distance from zero and is symmetric about the y-axis. This function is fundamental in understanding piecewise behavior and serves as a common example in exploring function transformations.

Horizontal Translation

Horizontal translation involves shifting a graph left or right by adjusting the input variable. Replacing x with (x + c) in a function translates the graph horizontally; specifically, a shift left by one unit is represented by replacing x with (x + 1).

Vertical Stretch

Vertical stretch refers to multiplying the function by a factor greater than 1, which causes the graph to stretch away from the x-axis. This transformation amplifies the output values while maintaining the graph's shape, establishing how heights and slopes are modified.

Vertical Translation

Vertical translation is the process of shifting the graph of a function upward or downward by adding or subtracting a constant from the function's output. Adding a constant to a function raises its entire graph by that constant value, altering the position without affecting the shape.

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