Plotting Function Transformations

Sometimes I come across things I cannot believe I didn’t know. That is one of the things I like about textbooks, since they give you a nice tour of the basics, they might not be complete, but you’re almost guaranteed that the material they have is worth learning.

Plotting basic mathematical transformations of functions can be quite useful. This plots show how adding a constant \(f(x+c)\) or multiplying \(f(cx)\) the argument of a function modifies the function:

color_2 <- "#ffffcc"

lbls <- c(expression(- pi),
          expression(-pi/2),
          0,
          expression(pi/ 2),
          expression(pi))

  
# Helper to return the function layer
make_fun_layer <- function(f) {
  
  color <- "#e57373"
  stat_function(fun = f, 
                      geom = "line", 
                      linetype = "dashed", 
                      size = 1,
                      colour = color)
}

basic_plot <- ggplot(tibble(x = c(-pi, pi)), aes(x = x)) +
  stat_function(fun = sin, 
                geom = "line", 
                size = 1,
                colour = "black") +
  scale_x_continuous(
    breaks = seq(- pi, pi, pi/2),
    labels = lbls
  ) + 
  geom_segment(aes(x=x, xend = xend, y=y, yend=yend), 
               size = 0.5,
               data=tibble(x=-pi, y=0, xend=pi, yend=0),
               arrow = arrow(length = unit(0.2, "cm"))) + 
  geom_segment(aes(x=0, xend=xend, y=y, yend=yend),
               size = 0.5,
               data=tibble(x=0, y=-1, yend=1.2, xend=0),
               lineend = "butt",
               arrow = arrow(length = unit(0.2, "cm"))) + 
  coord_fixed(ratio=1)+
  theme_blog + 
  theme(
    plot.title = element_text(hjust = 0.5),
    legend.title = element_blank(),
  )

basic_plot + 
  make_fun_layer(function(x) sin(x) + 1/2) +
  labs(title="sin(x) + c")

Now we can plot adding and multiplying to x and to y with a reusable function and display the 4 plots in a single pane with patchwork:

basic_plot + 
  make_fun_layer(function(x) sin(x + 1/2)) +
  labs(x="x", title="sin(x + c)")

basic_plot + 
  make_fun_layer(function(x) sin(2*x) ) +
  labs(x="x", title="sin(c*x)")

basic_plot + 
  make_fun_layer(function(x) 2 * sin(x)) +
  labs(x="x", title="c * sin(x)")