| Notes and Code |
Outcome |
|
Plotting a Math Function
First we graph equation (3)
$$ h(x) = \frac{x^2}{12} \tag{3} $$
by creating a loop to generate x values, and then using the equation to calculate the corresponding y values. We'll plot these points as spheres.
xaxis = curve(pos=[vec(-10,0,0), vec(10,0,0)])
yaxis = curve(pos=[vec(0,-10,0), vec(0,10,0)])
for x in range(-10, 10, 1):
y = x**2 / 12.0
sphere(pos=vec(x,y,0), radius=0.25)
Line 4: The for loop that creates values of x between -10 and 10 with a step of 1.
Lines 5: Calculate y using equation (3)
Line 6: Draw the sphere.
|
|
|
Connect the dots
We can draw line segments between the dots by creating a curve and appending points to the curve within the loop.
# create curve
line = curve()
for x in range(-10, 10, 1):
y = x**2 / 12
sphere(pos=vec(x,y,0), radius=0.25)
#append points to curve
line.append(pos=vec(x,y,0))
Line 2: create curve
Lines 8: Add points to the curve.
|
|
Creating a Function
Now, we're going to calculate the y-value for Equation 3
\[ h(x) = \frac{x^2}{12} \tag{3} \]
using a function. A function needs input/s and has output/s.
Input:
Our equation needs an x value.
Output/s:
The function returns a y value
The function is defined on Lines 1-3
Line 1:
- The
def keyword indicates that we're defining a function.
- the name of the function is
h, it can be pretty much anything, but adheres to the naming rules for variables (can't start with a number for example)
- the input of the function is listed within parentheses. The function recieves the input value and gives it the variable name
x_in for use within the function. This variable only exists within this function, making it a local variable, so I could have named it anything. In fact, it would probably have been better to name it simply x but I thought giving it a different name would clarify the difference between the value sent to the function (on line 12) and the variable name in the function.
Line 2:
- Here we calculate the y value of the function.
Line 3:
- Here we return the calculated value (
s) to the program.
- NOTE: Functions stand alone. When you run the program, the function definition is stored in memory and saved until the function is called.
Line 12:
- Calling the function. The function is called by using its name, and passing the input values. The function returns the result to the place in the code where the function is called.
|
def h(x_in):
y_out = x_in**2 / 12
return y_out
xaxis = curve(pos=[vec(-10,0,0), vec(10,0,0)])
yaxis = curve(pos=[vec(0,-10,0), vec(0,10,0)])
# create curve
line = curve()
for x in range(-10, 10, 1):
y = h(x)
sphere(pos=vec(x,y,0), radius=0.25)
#append points to curve
line.append(pos=vec(x,y,0))
|
Adding a second function
To demonstrate that you can use the same variable names within a function (so they're local variables) without affecting the variables outside the function, I'll add equation (1)
def f(x):
y = x - 4
return y
def h(x_in):
y_out = x_in**2 / 12
return y_out
xaxis = curve(pos=[vec(-10,0,0), vec(10,0,0)])
yaxis = curve(pos=[vec(0,-10,0), vec(0,10,0)])
# create curves
line = curve()
line1 = curve()
for x in range(-10, 10, 1):
y = h(x)
sphere(pos=vec(x,y,0), radius=0.25)
#append points to curve
line.append(pos=vec(x,y,0))
#draw equation 1
y = f(x)
sphere(pos=vec(x,y,0), radius=.25, color=color.red)
line1.append(pos=vec(x,y,0), color=color.red)
Now you should be able to add as many functions as you'd like (and reuse the same variable names).
|
|
|
Adding Functions
What happens if we add equations (1) and (3)
$$ k(x) = f(x) + h(x) $$
Show Notes ▼
def f(x):
y = x - 4
return y
def h(x_in):
y_out = x_in**2 / 12
return y_out
xaxis = curve(pos=[vec(-10,0,0), vec(10,0,0)])
yaxis = curve(pos=[vec(0,-10,0), vec(0,10,0)])
# create curves
line = curve()
line1 = curve()
line2 = curve()
for x in range(-10, 10, 1):
y = h(x)
sphere(pos=vec(x,y,0), radius=0.25)
#append points to curve
line.append(pos=vec(x,y,0))
#draw equation 1
y = f(x)
sphere(pos=vec(x,y,0), radius=.25, color=color.red)
line1.append(pos=vec(x,y,0), color=color.red)
# combine equations
y = f(x) + h(x)
sphere(pos=vec(x,y,0), radius=.25, color=color.green)
line2.append(pos=vec(x,y,0), color=color.green)
You can add, subtract, multiply, and divide functions in a similar way. And, you can use as many functions as you'd like.
|
|
|
Functions within Functions
What happens if we put equation (1) into equation (3):
\[ m(x) = f \circ h \]
which can also be written as:
\[ m(x) = f(h(x)) \]
|
|
A Function to Draw Functions: Passing a function to another function
You will have noticed that in the for loop where we're drawing the spheres and lines, we're repeating the same code for each function.
Whenever you see repetition in code it usually indicates that there's some way to simplify or combine things. So, we'll write a function that takes a mathematical function, and draws it.
def f(x):
y = x - 4
return y
def h(x_in):
y_out = x_in**2 / 12
return y_out
def drawFunction(f):
c = curve()
for x in range(-10,10,1):
y = f(x)
sphere(pos=vec(x,y,0), radius=0.25)
c.append(pos=vec(x,y,0))
xaxis = curve(pos=[vec(-10,0,0), vec(10,0,0)])
yaxis = curve(pos=[vec(0,-10,0), vec(0,10,0)])
# draw the functions:
drawFunction(f)
drawFunction(h)
Notice how much shorter the code is.
|
|
Default Values
Of course the drawFunction function only draws white points and curves. Let's leave that as the default, allow it to be changed. We'll do this by adding another input variable to the drawFunction function, but give it a default value (of white: color.white)
def f(x):
y = x - 4
return y
def h(x_in):
y_out = x_in**2 / 12
return y_out
def drawFunction(f, col=color.white):
c = curve()
for x in range(-10,10,1):
y = f(x)
sphere(pos=vec(x,y,0), radius=0.25, color=col)
c.append(pos=vec(x,y,0), color=col)
xaxis = curve(pos=[vec(-10,0,0), vec(10,0,0)])
yaxis = curve(pos=[vec(0,-10,0), vec(0,10,0)])
# draw the functions:
drawFunction(f)
drawFunction(h, color.red)
With a default value, we don't break our old code like:
drawFunction(f)
But can change any of the other function calls to give it a different color.
drawFunction(f, color.red)
|
|
Exercise
A. Create a function that graphs:
$$ n(x) = \left| \frac{x^2}{10} - 2 \right| \tag{4} $$
B. Change the drawFunction function so it takes a second, optional variable, that sets the step in the for loop to allow you to plot more points, and make smoother curves.
|
|
Exercise
B. Change the drawFunction function so it takes a second, optional variable, that sets the step in the for loop to allow you to plot more points, and make smoother curves.
Use this to graph equation (4):
$$ n(x) = \left| \frac{x^2}{10} - 2 \right| \tag{4} $$
Hint: to use non-integer steps, you'll have to use the arange function instead of range in your for loop.
|
|
