Math Problem Statement
Draw the graphs of ππ(π₯π₯)=3π₯π₯+2 and ππ(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 and on the same Cartesian plane. The point where they intersect is at .
Would you like further explanation on how to find the intersection or any other details?
Here are 5 related questions to explore:
- How can you determine the intersection point algebraically without graphing?
- What is the slope and y-intercept of each function, and what do they represent graphically?
- How does the slope of the lines affect their intersection?
- How would the graph change if the coefficients of or the constant terms were different?
- 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 and solve for 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
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