This Quadratic Regression Calculator helps you analyze data that follows a parabolic pattern. It fits data to the model, providing comprehensive analysis including model coefficients, goodness-of-fit statistics, extremum point detection, and diagnostic plots. This is useful for modeling data with U-shaped or inverted U-shaped relationships, such as optimal pricing, ballistic trajectories, and many physical processes. To learn about the data format required and test this calculator, .
Quadratic Regression models the relationship between a predictor variable (X) and a response variable (Y) as a parabola. The model has the form , where , , and are constants determined by the regression. When , the parabola opens upward; when , it opens downward.
Quadratic Model:
Vertex Formula (Minimum or Maximum Point):
Y-value at Vertex:
Derivative of the Quadratic Function:
| Time (hours) | Distance (km) |
|---|---|
| 0 | 0 |
| 1 | 25 |
| 2 | 40 |
| 3 | 45 |
| 4 | 40 |
| 5 | 25 |
After regression analysis, we get the equation:
The x-coordinate of the vertex is:
The y-coordinate of the vertex is:
The quadratic equation models the distance traveled by an object that initially accelerates and then decelerates.
The vertex at (3, 45) indicates that the maximum distance of 45 km is reached after 3 hours.
The negative coefficient of x² (-5) indicates a downward opening parabola, consistent with a maximum point.
library(tidyverse)
set.seed(42)
data <- tibble(
x = seq(-5, 5, by = 0.5),
y = 2*x^2 - 5*x + 3 + rnorm(21, 0, 5)
)
# Fit quadratic model
quad_model <- lm(y ~ x + I(x^2), data = data)
summary(quad_model)
# Compare with linear model
linear_model <- lm(y ~ x, data = data)
anova(linear_model, quad_model)
# Extract coefficients
a <- coef(quad_model)[3] # x^2 term
b <- coef(quad_model)[2] # x term
c <- coef(quad_model)[1] # Intercept
# Calculate vertex (minimum/maximum point)
x_vertex <- -b/(2*a)
y_vertex <- c + b*x_vertex + a*x_vertex^2
str_glue("Quadratic function: y = {a} x^2 + {b} x + {c}")
str_glue("Vertex point: ({x_vertex}, {y_vertex})")
str_glue("This is a {ifelse(a > 0, 'minimum', 'maximum')}")
# Plot the results
ggplot(data, aes(x = x, y = y)) +
geom_point() +
geom_smooth(method = "lm", formula = y ~ x + I(x^2), color = "red") +
geom_vline(xintercept = x_vertex, linetype = "dashed", color = "green") +
annotate("point", x = x_vertex, y = y_vertex, color = "green", size = 3) +
labs(title = "Quadratic Regression",
subtitle = str_glue("y = {round(a, 3)}x² + {round(b, 3)}x + {round(c, 3)}"),
x = "x",
y = "y") +
theme_minimal()import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import stats
import statsmodels.api as sm
np.random.seed(42)
x = np.linspace(-5, 5, 40)
y = 2*x**2 - 5*x + 3 + np.random.normal(0, 5, size=len(x))
data = pd.DataFrame({'x': x, 'y': y})
# Fit quadratic model
X = sm.add_constant(np.column_stack((x, x**2)))
model = sm.OLS(y, X).fit()
print(model.summary())
# Extract coefficients
c = model.params[0] # Intercept
b = model.params[1] # x coefficient
a = model.params[2] # x^2 coefficient
print(f"Quadratic equation: y = {a:.4f}x² + {b:.4f}x + {c:.4f}")
# Calculate vertex (minimum/maximum)
x_vertex = -b / (2*a)
y_vertex = c + b*x_vertex + a*x_vertex**2
print(f"Vertex point: ({x_vertex:.4f}, {y_vertex:.4f})")
print(f"This is a {'minimum' if a > 0 else 'maximum'}")
# Plot the results
plt.figure(figsize=(10, 6))
plt.scatter(x, y, alpha=0.7, label='Data')
# Plot regression line
x_sorted = np.sort(x)
X_sorted = sm.add_constant(np.column_stack((x_sorted, x_sorted**2)))
y_pred = model.predict(X_sorted)
plt.plot(x_sorted, y_pred, 'r-', label='Quadratic Fit')
# Mark vertex
plt.axvline(x=x_vertex, color='green', linestyle='--', alpha=0.5)
plt.plot(x_vertex, y_vertex, 'go', markersize=8, label=f"{'Minimum' if a > 0 else 'Maximum'} Point")
plt.title(f"Quadratic Regression: y = {a:.2f}x² + {b:.2f}x + {c:.2f}")
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()This Quadratic Regression Calculator helps you analyze data that follows a parabolic pattern. It fits data to the model, providing comprehensive analysis including model coefficients, goodness-of-fit statistics, extremum point detection, and diagnostic plots. This is useful for modeling data with U-shaped or inverted U-shaped relationships, such as optimal pricing, ballistic trajectories, and many physical processes. To learn about the data format required and test this calculator, .
Quadratic Regression models the relationship between a predictor variable (X) and a response variable (Y) as a parabola. The model has the form , where , , and are constants determined by the regression. When , the parabola opens upward; when , it opens downward.
Quadratic Model:
Vertex Formula (Minimum or Maximum Point):
Y-value at Vertex:
Derivative of the Quadratic Function:
| Time (hours) | Distance (km) |
|---|---|
| 0 | 0 |
| 1 | 25 |
| 2 | 40 |
| 3 | 45 |
| 4 | 40 |
| 5 | 25 |
After regression analysis, we get the equation:
The x-coordinate of the vertex is:
The y-coordinate of the vertex is:
The quadratic equation models the distance traveled by an object that initially accelerates and then decelerates.
The vertex at (3, 45) indicates that the maximum distance of 45 km is reached after 3 hours.
The negative coefficient of x² (-5) indicates a downward opening parabola, consistent with a maximum point.
library(tidyverse)
set.seed(42)
data <- tibble(
x = seq(-5, 5, by = 0.5),
y = 2*x^2 - 5*x + 3 + rnorm(21, 0, 5)
)
# Fit quadratic model
quad_model <- lm(y ~ x + I(x^2), data = data)
summary(quad_model)
# Compare with linear model
linear_model <- lm(y ~ x, data = data)
anova(linear_model, quad_model)
# Extract coefficients
a <- coef(quad_model)[3] # x^2 term
b <- coef(quad_model)[2] # x term
c <- coef(quad_model)[1] # Intercept
# Calculate vertex (minimum/maximum point)
x_vertex <- -b/(2*a)
y_vertex <- c + b*x_vertex + a*x_vertex^2
str_glue("Quadratic function: y = {a} x^2 + {b} x + {c}")
str_glue("Vertex point: ({x_vertex}, {y_vertex})")
str_glue("This is a {ifelse(a > 0, 'minimum', 'maximum')}")
# Plot the results
ggplot(data, aes(x = x, y = y)) +
geom_point() +
geom_smooth(method = "lm", formula = y ~ x + I(x^2), color = "red") +
geom_vline(xintercept = x_vertex, linetype = "dashed", color = "green") +
annotate("point", x = x_vertex, y = y_vertex, color = "green", size = 3) +
labs(title = "Quadratic Regression",
subtitle = str_glue("y = {round(a, 3)}x² + {round(b, 3)}x + {round(c, 3)}"),
x = "x",
y = "y") +
theme_minimal()import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import stats
import statsmodels.api as sm
np.random.seed(42)
x = np.linspace(-5, 5, 40)
y = 2*x**2 - 5*x + 3 + np.random.normal(0, 5, size=len(x))
data = pd.DataFrame({'x': x, 'y': y})
# Fit quadratic model
X = sm.add_constant(np.column_stack((x, x**2)))
model = sm.OLS(y, X).fit()
print(model.summary())
# Extract coefficients
c = model.params[0] # Intercept
b = model.params[1] # x coefficient
a = model.params[2] # x^2 coefficient
print(f"Quadratic equation: y = {a:.4f}x² + {b:.4f}x + {c:.4f}")
# Calculate vertex (minimum/maximum)
x_vertex = -b / (2*a)
y_vertex = c + b*x_vertex + a*x_vertex**2
print(f"Vertex point: ({x_vertex:.4f}, {y_vertex:.4f})")
print(f"This is a {'minimum' if a > 0 else 'maximum'}")
# Plot the results
plt.figure(figsize=(10, 6))
plt.scatter(x, y, alpha=0.7, label='Data')
# Plot regression line
x_sorted = np.sort(x)
X_sorted = sm.add_constant(np.column_stack((x_sorted, x_sorted**2)))
y_pred = model.predict(X_sorted)
plt.plot(x_sorted, y_pred, 'r-', label='Quadratic Fit')
# Mark vertex
plt.axvline(x=x_vertex, color='green', linestyle='--', alpha=0.5)
plt.plot(x_vertex, y_vertex, 'go', markersize=8, label=f"{'Minimum' if a > 0 else 'Maximum'} Point")
plt.title(f"Quadratic Regression: y = {a:.2f}x² + {b:.2f}x + {c:.2f}")
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()