Function Transformations Explained with Animations

Welcome to our exploration of function transformations! Today we'll learn how to shift functions horizontally and vertically.Let's start with our parent function, f of x equals x squared.Adding inside the parentheses shifts the graph left. Here, f of x plus 2 shifts the graph 2 units left.Let's summarize these important rules about function shifts.Let's try combining these shifts. For f of x plus 1 plus 1, the graph shifts left 1 unit and up 1 unit.When we multiply x by a constant before squaring, we affect the horizontal shape of the parabola.When we multiply x by 2, making the absolute value of a greater than 1, the graph compresses horizontally.When we multiply x by one-half, making the absolute value of a less than 1, the graph stretches horizontally.Now let's look at vertical transformations. When we multiply the entire function by 2, the graph stretches vertically.When we multiply by one-half, the graph compresses vertically.When we use negative values, interesting things happen. A negative value inside the function reflects the input before squaring.When we multiply the entire function by negative one, it reflects the graph across the x-axis.Let's summarize the key differences between horizontal and vertical transformations.Now that we understand individual transformations, let's combine them to create more complex functions.We'll start with our parent function, f of x equals x squared.First, we apply the horizontal shift. When we replace x with x minus 3, the graph shifts 3 units right.Next, we apply the vertical stretch by multiplying the function by 2. This doubles the height of every point on the graph.Finally, we add 1 to shift the entire graph up one unit.Let's look at another example that combines multiple transformations.This function includes a horizontal shift left, vertical compression, reflection, and vertical shift down.Let's review the key points about combining transformations.Thanks for learning about function transformations with Spark.E!