How do you plot a quadratic function in Python?

Use the equation y = ax² + bx + c to calculate y values for each x, then plot them using the Canvas API or SVG.

Can you animate a parabola in Python?

Yes. By updating x values over time and redrawing the curve, you can visualize parabolic motion or projectile paths dynamically.

What does the coefficient 'a' control in a quadratic function?

The coefficient 'a' determines the direction and width of the parabola. A positive 'a' opens upward, while a negative 'a' opens downward.

Solving and Animating a Quadratic Equation in Python

Understanding the Quadratic Function | Maths Explanation for Python Kids

In this tutorial, we'll explore how to solve quadratic equations in Python and use them to plot and animate a quadratic curve on an Turtle canvas. Understanding the quadratic equation in programming helps you create parabolic motion, simulations, and engaging math visuals.

A quadratic equation has the general form y = ax² + bx + c; where a, b, and c are constants. This tutorial explains how to solve, plot, and animate a quadratic function using Python and the Turtle canvas.

Generating Quadratic Curves for Python

To generate a quadratic curve in Python, you need two points - the starting point and the vertex (turning point - maximum or minimum).

Figure: Quadratic equation graph in Python showing quadratic points - start point and turning (maximum) point.

The general quadratic function is:

y = ax² + bx + c

^dy/_dx = y^I = 2ax + b
At maximum / minimum point, y^I = 0
y^I|_{(x = x_max)} = 0
2ax_max + b = 0
b = -2ax_max

Substituting b in the general equation
y = ax² + bx + c
  = ax² - 2ax_maxx + c

         At (x_start, y_start):
y_start = ax_start² - 2ax_maxx_start + c
         At (x_max, y_max):
y_max = ax_max² - 2ax_max² + c
    = -ax_max² + c     --------- (eqn *)

Subtracting both derived equations
y_start - y_max = ax_start² - 2ax_maxx_start + ax_max²
(x_start² - 2x_maxx_start + x_max²)a = y_start - y_max

Hence:

a =	y_start - y_max	=	y_start - y_max
	x_start² - 2x_maxx_start + x_max²		(x_start - x_max)²

b = -2ax_max
& from (eqn *)
c = y_max + ax_max²

Python Code Example: Quadratic Path Animation

Once we have the equation, we can generate a quadratic curve with Python to visualize its motion. The following example demonstrates how to animate an object along a quadratic curve in Python using the Turtle canvas. This is a simple form of quadratic motion simulation that helps visualize parabolic motion, such as a ball being thrown.

To make a body travel by the equation of a quadratic curve, continuously increment x by some interval, and use the quadratic equation to get the corresponding y value.

Create 2 new Python files; File, New File.
Call them Facet.py and QuadraticPath.py.
Type out the adjoining Python / Turtle code for animating an image body through the path of a quadratic curve.
This simple example demonstrates Python quadratic animation.

Important: When trying to click on the button to get things started, you might need to click away from the button text.

Key Takeaways on Quadratic Path Animation in Python

A quadratic function in Python models parabolic motion or curves.
The quadratic equation Python code can be used for plotting and animations.
The constants a, b, and c control the shape and direction of the parabola.

Applications in Programming and Education

The quadratic equation in Python is useful for programming concepts like projectile motion, trajectory planning, and smooth animation curves. Teachers can use this example to show how maths and coding connect - making parabolas come alive through Python programming.

Teachers and students can use this Python quadratic formula tutorial to explore math and programming together. It's a practical example of using maths with code.

Summary: Visualizing Quadratic Equations in Python

In this tutorial, you've learnt how to solve quadratic equations in Python using the quadratic formula. We've also explore how to plot and animate a quadratic curve on an Turtle canvas. Understanding how to code the quadratic equation in Python is useful for creating simulations, parabolic motion, and smooth animations.

By combining mathematics and Python, you can solve and animate quadratic equations easily. Whether you're plotting parabolas, simulating motion, or building educational tools, mastering the quadratic formula in Python gives you a solid foundation in computational math.

So! Python Fun Practice Exercise - Animate along Quadratic Path

As a fun practice exercise, try adjusting the coefficients a, b, and c to change the curve's shape or motion pattern. This will be a great way to connect mathematics and programming, and help you understand more about Python animations and quadratic equations.

Python Quadratic Path Code for Turtle Template - Facet Class

import turtle

class Template:
    def __init__(self):
        screen = turtle.getscreen()
        screen.title("usingMaths.com")
        screen.setup(width=0.9, height=0.9, startx=None, starty=None)
        screen.colormode(255)
        self.bgcolour = "#cccccc"
        screen.bgcolor(self.bgcolour)
        screen.delay(0)
        self.screen = screen

        self.wnd_width = self.screen.window_width()
        self.wnd_height = self.screen.window_height()

    def controlButtons(self):
        self.button = turtle.clone()

        turtle.speed(0)
        turtle.penup()
        turtle.setposition(-self.wnd_width/2, self.wnd_height/2)
        turtle.color("black", "pink")

        turtle.pendown()
        # draw button region
        turtle.begin_fill()
        turtle.forward(self.wnd_width)
        turtle.right(90)
        turtle.forward(50)
        turtle.right(90)
        turtle.forward(self.wnd_width)
        turtle.right(90)
        turtle.forward(50)
        turtle.end_fill()

        self.button.speed(0)
        self.button.penup()
        # draw button
        self.button.setposition(0, self.wnd_height/2 - 25)
        self.button.shape("square")
        self.button.resizemode("user")
        self.button.shapesize(2, 8)
        self.button.color("black", (255, 0, 255))

        return turtle.getturtle()

Python Animation Code for Quadratic Path file

#!usr/bin/env python3

import math
from Facet import Template

def prep():
    global x_start, y_start, x_max, y_max, x_dot, y_dot, a, b, c, restart
    restart = False

    # bind fun function
    scene.button.onrelease(play)
    screen.delay(20)

    turtle.penup()
    # button text
    turtle.setposition(scene.button.xcor(), scene.button.ycor()-10)
    turtle.pendown()
    turtle.write("Move", align="center", font=("Arial",16,"bold"))

    # dot parameters
    diameter = 1
    dot_colour = "#ffff00"
    x_start = 100 - scene.wnd_width/2; y_start = -scene.wnd_height/3
    x_max = 0; y_max = scene.wnd_height/3
    x_dot = x_start; y_dot = y_start

    # constants
    a = (y_start - y_max) / math.pow((x_start - x_max), 2)
    b = -2*a*x_max
    c = y_max + a*math.pow(x_max, 2)

    # transform turtle into a dot
    turtle.penup()
    turtle.setposition(x_dot, y_dot)
    turtle.setheading(0)
    turtle.shape("triangle")
    turtle.shapesize(diameter, diameter)
    turtle.color(dot_colour, dot_colour)

# fun function when button is clicked
# just moves turtle until it hits the right boundary
def play(x, y):
    global x_start, y_start, x_max, y_max, x_dot, y_dot, a, b, c, restart

    if restart:
        turtle.clearstamps()
        restart = False

    if x_dot <= 10*turtle.shapesize()[1] - x_start:
        y_dot = a*x_dot*x_dot + b*x_dot + c

        # change dot's coordinates
        turtle.setposition(x_dot, y_dot)
        turtle.stamp()

        x_dot += 20
        screen.ontimer(play(0,0), 500)
    else:
        x_dot = x_start; y_dot = y_start
        restart = True
        screen.mainloop()

# Let fly
scene = Template();
turtle = scene.controlButtons()
screen = scene.screen

prep()

I have a better version / See other versions.