Math Problem Statement

Find roots by graphing: f(x)=x^2-8x+12

Solution

To find the roots of f(x)=x28x+12f(x) = x^2 - 8x + 12 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 f(x)=0f(x) = 0.

Step 1: Rewrite the Function

The function is: f(x)=x28x+12f(x) = x^2 - 8x + 12

Step 2: Plot the Graph

The equation f(x)=x28x+12f(x) = x^2 - 8x + 12 represents a parabola that opens upwards because the coefficient of x2x^2 is positive.

  1. Find the vertex: The x-coordinate of the vertex is given by: x=b2ax = -\frac{b}{2a} Here, a=1a = 1, b=8b = -8, and c=12c = 12. Substituting: x=82(1)=82=4x = -\frac{-8}{2(1)} = \frac{8}{2} = 4 The y-coordinate is found by substituting x=4x = 4 into f(x)f(x): f(4)=428(4)+12=1632+12=4f(4) = 4^2 - 8(4) + 12 = 16 - 32 + 12 = -4 So, the vertex is (4,4)(4, -4).

  2. Find the roots (x-intercepts): Set f(x)=0f(x) = 0: x28x+12=0x^2 - 8x + 12 = 0 Factoring: (x6)(x2)=0(x - 6)(x - 2) = 0 Thus, x=6x = 6 and x=2x = 2.

  3. Plot the points:

    • The vertex is (4,4)(4, -4).
    • The roots are (6,0)(6, 0) and (2,0)(2, 0).
    • The y-intercept is (0,12)(0, 12) (when x=0x = 0).

Step 3: Graph

The graph will show a parabola passing through the roots x=6x = 6 and x=2x = 2, with the vertex at (4,4)(4, -4).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 f(x)=x28x+12f(x) = x^2 - 8x + 12. The roots are at x=2x = 2 and x=6x = 6, where the parabola crosses the x-axis. The vertex is at (4,4)(4, -4), and the y-intercept is at (0,12)(0, 12).

Do you have any questions or want further details?

Here are 5 related questions for further exploration:

  1. How can the vertex form of the equation help in understanding the graph?
  2. What does the discriminant tell us about the roots of this quadratic equation?
  3. How do the coefficients aa, bb, and cc affect the shape and position of the parabola?
  4. What are other methods to find the roots besides graphing?
  5. 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.

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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