Math Problem Statement

Draw the graphs of 𝑓𝑓(π‘₯π‘₯)=3π‘₯π‘₯+2 and 𝑔𝑔(π‘₯π‘₯)=2π‘₯π‘₯+3 on the same cartesian plane. Show the point of intersection if the two functions do intersect.

Solution

numpy as np import matplotlib.pyplot as plt

Define the functions

def f(x): return 3*x + 2

def g(x): return 2*x + 3

Create an array of x values

x = np.linspace(-10, 10, 400)

Calculate the y values for each function

y_f = f(x) y_g = g(x)

Create the plot

plt.figure(figsize=(8, 6)) plt.plot(x, y_f, label=r'$f(x) = 3x + 2$', color='blue') plt.plot(x, y_g, label=r'$g(x) = 2x + 3$', color='red')

Find the point of intersection by solving 3x + 2 = 2x + 3

3x - 2x = 3 - 2 -> x = 1

Substituting x = 1 into f(x) or g(x), y = 5

x_intersection = 1 y_intersection = f(x_intersection)

Plot the intersection point

plt.scatter(x_intersection, y_intersection, color='green', zorder=5) plt.text(x_intersection, y_intersection, f'({x_intersection}, {y_intersection})', color='green', fontsize=12, ha='left')

Label the axes and the graph

plt.axhline(0, color='black',linewidth=1) plt.axvline(0, color='black',linewidth=1) plt.grid(True) plt.title("Graphs of $f(x) = 3x + 2$ and $g(x) = 2x + 3$") plt.xlabel('x') plt.ylabel('y') plt.legend()

Show the plot

plt.show()Here is the graph of the two functions f(x)=3x+2f(x) = 3x + 2 and g(x)=2x+3g(x) = 2x + 3 on the same Cartesian plane. The point where they intersect is at (1,5)(1, 5).

Would you like further explanation on how to find the intersection or any other details?

Here are 5 related questions to explore:

  1. How can you determine the intersection point algebraically without graphing?
  2. What is the slope and y-intercept of each function, and what do they represent graphically?
  3. How does the slope of the lines affect their intersection?
  4. How would the graph change if the coefficients of xx or the constant terms were different?
  5. What would happen if the two functions were parallel?

Tip: When two linear functions intersect, the solution is simply the point where their equations are equal. Set f(x)=g(x)f(x) = g(x) and solve for xx to find the intersection.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Graphing
Intersection of Functions
Algebra

Formulas

f(x) = 3x + 2
g(x) = 2x + 3

Theorems

Equating Linear Functions to Find Intersection

Suitable Grade Level

Grades 7-9