Vertical and horizontal and reflections and compressions change the change the shape of the graph.

Yes, vertical and horizontal transformations, as well as reflections and compressions, significantly alter the shape and position of a graph. Here's a breakdown of each type of transformation: ### 1. Vertical Transformations: - **Vertical Shifts**: Adding or subtracting a constant \( k \) to a function \( f(x) \) results in a vertical shift. For example, \( f(x) + k \) shifts the graph up by \( k \) units, while \( f(x) - k \) shifts it down by \( k \) units. - **Vertical Stretch/Compression**: Multiplying the function by a constant \( a \) (where \( |a| > 1 \) for stretching and \( 0 < |a| < 1 \) for compressing) changes the vertical scale of the graph. For example, \( a \cdot f(x) \) stretches the graph vertically if \( a > 1 \) and compresses it if \( 0 < a < 1 \). ### 2. Horizontal Transformations: - **Horizontal Shifts**: Adding or subtracting a constant \( h \) inside the function argument results in a horizontal shift. For example, \( f(x - h) \) shifts the graph to the right by \( h \) units, while \( f(x + h) \) shifts it to the left by \( h \) units. - **Horizontal Stretch/Compression**: Multiplying the input \( x \) by a constant \( b \) (where \( |b| > 1 \) for compression and \( 0 < |b| < 1 \) for stretching) changes the horizontal scale. For example, \( f(bx) \) compresses the graph horizontally if \( |b| > 1 \) and stretches it if \( 0 < |b| < 1 \). ### 3. Reflections: - **Reflection Across the x-axis**: Multiplying the function by -1, i.e., \( -f(x) \), reflects the graph across the x-axis. - **Reflection Across the y-axis**: Replacing \( x \) with \( -x \), i.e., \( f(-x) \), reflects the graph across the y-axis. ### Summary: These transformations can be combined to create more complex changes to the graph. For example, the function \( a \cdot f(b(x - h)) + k \) involves a vertical stretch/compression by \( a \), a horizontal compression/stretch by \( b \), a horizontal shift by \( h \), and a vertical shift by \( k \). Understanding these transformations helps in sketching graphs and analyzing their behavior.