The expression 'a*f(x)' represents a vertical scaling of the function f(x) by the factor 'a'. This transformation affects the amplitude of the graph, stretching it if |a| > 1 and compressing it if 0 < |a| < 1. The sign of 'a' also influences the orientation of the graph, reflecting it over the x-axis when 'a' is negative.
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'a' can be any real number, and its absolute value determines how much stretching or compressing occurs.
If 'a' is positive, the graph retains its original orientation, while a negative 'a' will flip the graph over the x-axis.
Vertical scaling does not affect the x-intercepts of the function; they remain unchanged regardless of the value of 'a'.
The y-intercepts of the function are affected by 'a'; if you change 'a', you will change the height at which the graph intersects the y-axis.
For periodic functions like sine and cosine, changing 'a' alters their amplitude, impacting how high and low the peaks and valleys appear.
Review Questions
How does changing the value of 'a' in 'a*f(x)' affect the shape and position of the graph?
'a' influences both the amplitude and orientation of the graph. If |a| > 1, the graph undergoes a vertical stretch, making it taller. If 0 < |a| < 1, it undergoes a vertical compression, making it shorter. Additionally, if 'a' is negative, it reflects the graph over the x-axis. This means that points on f(x) move either further away from or closer to the x-axis depending on whether 'a' is greater than or less than 1.
Explain how vertical scaling with 'a*f(x)' maintains or changes x-intercepts and y-intercepts of a function.
Vertical scaling affects y-intercepts but does not change x-intercepts. Since x-intercepts are determined by setting f(x) to zero, they remain unchanged regardless of how 'a' scales f(x). However, y-intercepts do change; they get scaled by 'a', meaning if 'f(0)' is your original y-intercept, then your new y-intercept will be 'a*f(0)'. This differentiation shows how scaling modifies vertical positions without altering horizontal intercepts.
Analyze how vertical transformations, such as those represented by 'a*f(x)', impact periodic functions and their characteristics.
For periodic functions like sine or cosine, modifying 'a' directly alters their amplitude while keeping their period constant. When 'a' increases (|a| > 1), peaks rise higher and valleys drop lower, enhancing oscillation intensity. Conversely, reducing 'a' (0 < |a| < 1) minimizes these peaks and valleys, making them closer to the midline. The effect is particularly significant in applications like sound waves or tides where amplitude translates to volume or height, demonstrating real-world relevance in understanding periodic behavior through these transformations.
A transformation that increases the distance between points on the graph of a function away from the x-axis, occurring when |a| > 1.
Vertical Compression: A transformation that decreases the distance between points on the graph of a function towards the x-axis, occurring when 0 < |a| < 1.