For the following exercises, write an equation for each graphed function by using transformation

step 1 Okay, so this graph on the left over here is going to be our parent function, which is the squar

For the following exercises, write an equation for each...

Key Concepts

Function Transformations

Function transformations involve modifying a basic or 'parent' function to produce a new function. These modifications can include translating (shifting), scaling (stretching or compressing), reflecting, or a combination of these changes. Understanding these alterations allows you to write equations that represent the transformed graph based on its original form.

Parent Functions

Parent functions, sometimes called toolkit functions, are fundamental functions that serve as the building blocks for more complex behaviors. By applying transformations to these simple functions, one can create a wide variety of graphs. Recognizing the structure of parent functions is essential for determining the correct alterations needed to write the equation of a transformed graph.

Translation

Translation is a type of function transformation where the graph is shifted horizontally, vertically, or both, without changing its shape or orientation. Horizontal translations move the graph left or right, while vertical translations move it up or down. These shifts are added to the function equation as constants modifying the variable or the output.

Scaling (Stretching/Compression)

Scaling transformations change the size of the graph either vertically or horizontally. Vertical scaling involves multiplying the output of the function to stretch or compress the graph along the y-axis, while horizontal scaling involves multiplying the input to affect the x-axis. This transformation alters the steepness or width of the graph without changing its overall structure.

Reflection

Reflection transforms the graph of a function by flipping it over a specific axis, most commonly the x-axis or y-axis. Reflection over the x-axis involves multiplying the entire function by -1, while reflection over the y-axis involves replacing the variable with its negative. This operation is useful for understanding the symmetry properties of functions and writing equations that mirror a given graph.

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