Play around with the various constants to try to get a sense of what they do. Be sure to change the base function to make sure your hypotheses hold for other base functions as well.
In particular, try adjusting the second and third constants (inside the parentheses) at the same time. Does the graph stretch before shifting or after? Hint: it may help to check the "draw guidelines" checkbox.
Try adjusting all of the constants, and checking "draw key points" and "label key points". Verify that the moved key point falls on the graph, by plugging it into the equation. What number did you wind up having to plug into the base function?
Check "show original graph" and try to find different constants that give you the same graph, or nearly the same graph. Try this for various base functions. Once you find one, try simplifying the equation for the new graph.
Play around with the case where the base function is x. How does this relate to what we learned when studying lines?
Try reducing the graph resolution. See if you can figure out how this function grapher works.
Hint: You can copy the right hand side of the EQUATION below and paste it into textboxes in the page, and in the Piecewise Functions interactive.
Here you can play around with the constants a, b, c and d, and choose the function f, and see the graph of a*f(b*x + c) + d:
Base Function:
a:
b:
c:
d:
EQUATION:
Min x:
Max x:
Min y:
Max y:
Draw key points
Label key points (doesn't do anything unless key points are drawn)
Show original graph
Draw axes
Draw grid
Draw guidelines
Graph resolution:
(number of points. lower values are faster, larger values will smooth graph, reduce missing pieces of graph of sqrt(x))
Stretch a cat:
The cat below consists of a series of points connected by lines. The x and y coordinates of these points are multiplied by the constants you specify.
How are stretching, squashing, and mirroring related?
Horizontal:
Vertical:
Guessing Game:
The interactive below will give you the graph of a function and ask you to guess a formula for the graph. Note that there may be many solutions. Guesses are interpreted in the javascript interpreter. This means:
You will need to use notation similar to that in the interactive at the top of the page.
Specifically, you will need to use ** for exponentiation. Use x**2, not x^2.
You will also need to use * for multiplication. Javascript does not automatically interpret parentheses or juxtaposition as multiplication. Use 2*(x**2) or 2*x**2, not 2x**2 or 2(x**2).
If your function doesn't appear to work, you may want to try adding parentheses. Use -(x**2), not -x**2.
Using variables used in the code of this webpage may give bad results.
Please avoid entering arbitrary javascript code as this will execute it many times.
Uncheck to use as graphing calculator
Check to increase difficulty
Show a possible solution
Min x:
Max x:
Min y:
Max y:
Graph resolution:
Play elsewhere:
Check out this incorrectly graphed function on Wolfram Alpha.
You can use calculus to show that the graph should be a smooth arc with a hole at x = 0.
Check out this incorrectly graphed function on Desmos. . You already know enough to figure out that the graph of this function should look like the graph of 1/(x+1), with a hole at x = -2.
Desmos (You'll need to click the circle in row 16 on the left to get this to work. Replace the letter f by some other letter to change the function.)
Geogebra Lets you play with the stretch/shift constants, allows you to enter your own base functions.
Geogebra Lets you play with the stretch/shift constants.