Math Problem Statement
Find roots by graphing: f(x)=x^2-8x+12
Solution
To find the roots of by graphing, we need to determine where the parabola crosses the x-axis. These points represent the roots of the equation, which are the solutions to .
Step 1: Rewrite the Function
The function is:
Step 2: Plot the Graph
The equation represents a parabola that opens upwards because the coefficient of is positive.
-
Find the vertex: The x-coordinate of the vertex is given by: Here, , , and . Substituting: The y-coordinate is found by substituting into : So, the vertex is .
-
Find the roots (x-intercepts): Set : Factoring: Thus, and .
-
Plot the points:
- The vertex is .
- The roots are and .
- The y-intercept is (when ).
Step 3: Graph
The graph will show a parabola passing through the roots and , with the vertex at .import numpy as np import matplotlib.pyplot as plt
Define the function
def f(x): return x**2 - 8*x + 12
Generate x values
x = np.linspace(0, 8, 400) y = f(x)
Plot the function
plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r"$f(x) = x^2 - 8x + 12$")
Mark the roots and vertex
roots = [2, 6] vertex = (4, -4) plt.scatter(roots, [0, 0], color='red', label="Roots") plt.scatter(*vertex, color='blue', label="Vertex (4, -4)")
Mark the y-intercept
y_intercept = (0, f(0)) plt.scatter(*y_intercept, color='green', label="y-intercept (0, 12)")
Add labels and grid
plt.axhline(0, color='black', linewidth=0.8, linestyle='--') plt.axvline(0, color='black', linewidth=0.8, linestyle='--') plt.title("Graph of $f(x) = x^2 - 8x + 12$") plt.xlabel("x") plt.ylabel("f(x)") plt.legend() plt.grid()
Show plot
plt.show() <<ImageDisplayed>>successThe graph above represents the function . The roots are at and , where the parabola crosses the x-axis. The vertex is at , and the y-intercept is at .
Do you have any questions or want further details?
Here are 5 related questions for further exploration:
- How can the vertex form of the equation help in understanding the graph?
- What does the discriminant tell us about the roots of this quadratic equation?
- How do the coefficients , , and affect the shape and position of the parabola?
- What are other methods to find the roots besides graphing?
- How does completing the square relate to finding the vertex?
Tip: Factoring is a quick way to find the roots of a quadratic equation when it is easily factorable.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Graphing Parabolas
Formulas
f(x) = ax^2 + bx + c
Vertex formula: x = -b / 2a
Factoring: (x - p)(x - q) = 0
Theorems
Quadratic Formula
Fundamentals of Parabolas
Suitable Grade Level
Grades 8-10