Describe in detail the four ways in which you can translate/move a function. Be specific and give examples.
Translating or moving a function involves shifting its graph horizontally or vertically on the coordinate plane. There are four primary ways in which a function can be translated/moved:
1. Horizontal translation: This involves shifting the function horizontally to the left or right. To translate a function horizontally, you add or subtract a value inside the parentheses of the function equation.
Example: Suppose we have the function f(x) = x^2. To translate it to the right by 2 units, we can modify the equation as f(x) = (x - 2)^2. In this case, every point on the graph of f(x) is shifted to the right by 2 units.
Similarly, to translate the function to the left, we would modify the equation as f(x) = (x + 2)^2, where every point on the graph of f(x) is shifted to the left by 2 units.
2. Vertical translation: This involves shifting the function vertically up or down. To translate a function vertically, you add or subtract a value outside the function.
Example: Consider the function f(x) = x^2. To translate it upward by 3 units, the equation becomes f(x) = x^2 + 3. In this case, every point on the graph of f(x) is shifted upward by 3 units.
Likewise, to translate the function downward, we modify the equation as f(x) = x^2 - 3, where every point on the graph of f(x) is shifted downward by 3 units.
3. Horizontal scaling: This involves stretching or compressing the function horizontally. To stretch or compress a function horizontally, you multiply or divide the variable (x) inside the function equation.
Example: Let's consider the function f(x) = x^2. To horizontally stretch the function by a factor of 2, we modify the equation as f(x) = (2x)^2. In this case, every x-coordinate on the graph is multiplied by 2, resulting in a flatter, wider parabola.
Similarly, to compress the function horizontally by a factor of 2, the equation becomes f(x) = (x/2)^2, where every x-coordinate is divided by 2, resulting in a steeper, narrower parabola.
4. Vertical scaling: This involves stretching or compressing the function vertically. To vertically stretch or compress a function, you multiply or divide the entire function equation.
Example: Suppose we have the function f(x) = x^2. To vertically stretch the function by a factor of 3, we can modify the equation as f(x) = 3(x^2). In this case, the y-coordinate of every point on the graph is multiplied by 3, resulting in a steeper, higher parabola.
Similarly, to vertically compress the function by a factor of 1/2, the equation becomes f(x) = (1/2)(x^2), where the y-coordinate is divided by 2, resulting in a flatter, lower parabola.
In summary, the four ways to translate/move a function are horizontal translation (left or right), vertical translation (up or down), horizontal scaling (stretch or compress), and vertical scaling (stretch or compress). By modifying the function equation, you can achieve various transformations of the graph.