Transformations of Functions and Straight Line Graphs

Transformations of Functions and Straight Line Graphs

Transformations of functions and their corresponding straight line graphs are fundamental concepts in understanding algebraic relationships. By manipulating functions through translations, reflections, and scalings, we can create a rich variety of new graphs to explore and analyze.

Transformations of Functions

1. Translation: A vertical shift moves the graph up or down by a fixed amount. A horizontal shift moves the graph left or right by a fixed amount. These shifts are represented by (f(x) = g(x-h)) for a horizontal shift by (h) and (f(x) = g(x+k)) for a vertical shift by (k).

2. Reflection: Reflecting a function over the (x)-axis or (y)-axis changes the graph's orientation. A reflection over the (x)-axis is indicated by changing the sign of the function's output ((f(x) = -g(x))). A reflection over the (y)-axis is indicated by changing the sign of the function's input ((f(x) = g(-x))).

3. Scaling: A function can be stretched or compressed horizontally or vertically. A horizontal scaling by a factor of (a) is indicated by ((f(x) = g(ax))). A vertical scaling by a factor of (b) is indicated by ((f(x) = bg(x))).

Interpreting Graphs

To understand the geometric relationships between transformed graphs, we analyze the intercepts and the slopes of the lines.

1. Intercepts: The (x)-intercept is where the graph crosses the (x)-axis, which is the point at which (y = 0). The (y)-intercept is where the graph crosses the (y)-axis, which is the point at which (x = 0).

2. Slope: The slope of a line is the ratio of the vertical change to the horizontal change between any two points on the line. The slope-intercept form of a line is (y = mx + b) where (m) is the slope and (b) is the (y)-intercept.

Finding Equations of Lines

To write the equation of a line, we can use either the slope-intercept form or the point-slope form.

1. Slope-intercept form: This form is derived from the slope-rise and run of the line. The slope-intercept form is (y = mx + b), where (m) is the slope and (b) is the (y)-intercept.

2. Point-slope form: This form is derived from the slope of the line and a specific point on the line ((x_1, y_1)). The point-slope form is (y - y_1 = m(x - x_1)).

By understanding these concepts, you'll be able to analyze, sketch, and write the equations of transformed functions and their corresponding straight line graphs.