Final answer:
The function rule for a vertical stretching of f(x) = |x| by a scale factor of 7 is g(x) = 7|x|. This operation multiplies each y-value by 7, causing the V-shaped graph of the absolute value function to become taller while keeping its vertex at the origin.
Step-by-step explanation:
To determine the function rule for a vertical stretching of the function f(x) = |x| by a scale factor of 7, you multiply the original function by 7 to scale the values appropriately. Hence, the new function after the stretch is expressed as g(x) = 7|x|. Vertical stretching of a graph means that each y-coordinate is multiplied by the scale factor, causing the graph to stretch away from the x-axis if the scale factor is greater than 1, or to compress towards the x-axis if it is between 0 and 1.
The absolute value function f(x) = |x| has a V-shape with its vertex at the origin (0,0). When applying the vertical stretch by a factor of 7, this V-shape becomes taller and narrower while maintaining the overall shape and symmetry. The vertex remains at the origin, but now each point on f(x) is seven times further from the x-axis. Therefore, the point (1, 1) on f(x) becomes (1, 7) on g(x), and similarly, the point (-1, 1) on f(x) becomes (-1, 7) on g(x). This expanded shape is still symmetric about the y-axis, maintaining the property of the original absolute value function being an even function.