Describe how the graph of y= absolute value changes when the equation is changed to y = -5 absolute value x-9 and + 12
To understand how the graph of y = absolute value changes when the equation is modified to y = -5| x-9 | + 12, we need to analyze each modification separately.
1. Scaling: The coefficient -5 in front of the absolute value function indicates a vertical scaling. A positive value for the coefficient would vertically stretch the graph, while a negative value reflects it vertically and stretches it in the opposite direction. In this case, -5 represents a vertical reflection and compression (by a factor of 5) of the graph.
2. Horizontal translation: The term (x - 9) inside the absolute value function represents a horizontal translation. The translation is in the positive direction since the equation is (x - 9). It means that the graph is shifted nine units to the right compared to the standard absolute value function y = |x|.
3. Vertical shift: The constant term 12 is responsible for the vertical shift of the graph. A positive value for the constant would shift the graph upward, while a negative value would shift it downward. In this case, 12 indicates that the graph is shifted twelve units up.
Combining these modifications, the graph of y = -5| x-9 | + 12 will be a vertically reflected, compressed graph of the standard absolute value function y = |x|, shifted nine units to the right, and twelve units up.
To visualize this transformation, you can plot the original absolute value function y = |x| on a graph and apply each modification step by step. Start by plotting key points on the original graph and then apply the transformations to each point individually.