✨ Math Magic - Complete Edition ✨

All Problems Solved with Interactive Graphs!

Exercise 7.9 - Part 1: Finding Asymptotes

Discover the hidden lines that graphs approach but never touch!

(i) \( f(x) = \frac{x^2}{x^2 - 1} \)

Find vertical asymptotes by setting denominator to zero: \( x^2 - 1 = 0 \)
Solve: \( x^2 = 1 \) → \( x = \pm 1 \)
Find horizontal asymptote by comparing degrees (both degree 2):
Divide leading coefficients: \( \frac{1}{1} = 1 \) → \( y = 1 \)
No oblique asymptotes (degree numerator ≤ degree denominator)

(ii) \( f(x) = \frac{x^2}{x + 1} \)

Vertical asymptote: \( x + 1 = 0 \) → \( x = -1 \)
No horizontal asymptote (numerator degree > denominator degree)
Find oblique asymptote by polynomial division:
\( \frac{x^2}{x + 1} = x - 1 + \frac{1}{x+1} \)
Oblique asymptote: \( y = x - 1 \)

(iii) \( f(x) = \frac{3x}{\sqrt{x^2 + 2}} \)

No vertical asymptotes (denominator never zero)
Find horizontal asymptotes as \( x \to \pm\infty \):
For \( x \to \infty \): \( \frac{3x}{\sqrt{x^2}} = \frac{3x}{|x|} = 3 \)
For \( x \to -\infty \): \( \frac{3x}{\sqrt{x^2}} = \frac{3x}{|x|} = -3 \)
Horizontal asymptotes: \( y = 3 \) (right), \( y = -3 \) (left)

(iv) \( f(x) = \frac{x^2 - 6x - 1}{x + 3} \)

Vertical asymptote: \( x + 3 = 0 \) → \( x = -3 \)
No horizontal asymptote (numerator degree > denominator degree)
Find oblique asymptote by polynomial division:
\( \frac{x^2 - 6x - 1}{x + 3} = x - 9 + \frac{26}{x+3} \)
Oblique asymptote: \( y = x - 9 \)

(v) \( f(x) = \frac{x^2 + 6x - 4}{3x - 6} \)

Vertical asymptote: \( 3x - 6 = 0 \) → \( x = 2 \)
No horizontal asymptote (numerator degree > denominator degree)
Find oblique asymptote by polynomial division:
\( \frac{x^2 + 6x - 4}{3x - 6} = \frac{1}{3}x + \frac{8}{3} + \frac{12}{3x-6} \)
Oblique asymptote: \( y = \frac{1}{3}x + \frac{8}{3} \)
Did You Know? The word "asymptote" comes from the Greek "asymptotos" meaning "not falling together" - describing how the curve approaches but never quite reaches the line!

Exercise 7.9 - Part 2: Sketching Graphs

Visualize these functions with all their key features!

(i) \( y = -\frac{1}{3}(x^3 - 3x + 2) \)

This is a cubic function (degree 3)
Find roots: \( x^3 - 3x + 2 = 0 \) → \( (x-1)^2(x+2) = 0 \)
Roots at \( x = 1 \) (double root) and \( x = -2 \)
No asymptotes (polynomial function)
End behavior: As \( x \to \infty \), \( y \to -\infty \); as \( x \to -\infty \), \( y \to \infty \)

(ii) \( y = x\sqrt{4-x} \)

Domain: \( 4 - x \geq 0 \) → \( x \leq 4 \)
x-intercepts: \( x = 0 \) and \( x = 4 \)
No asymptotes (not a rational function)
Behavior: Curve starts at (-∞, -∞) and ends at (4, 0)
Maximum point at x ≈ 2.667 (found using calculus)

(iii) \( y = \frac{x^2 + 1}{x^2 - 4} \)

Vertical asymptotes: \( x^2 - 4 = 0 \) → \( x = \pm 2 \)
Horizontal asymptote: \( y = 1 \) (same degree)
y-intercept: \( y = \frac{1}{-4} = -0.25 \)
No x-intercepts (numerator always positive)

(iv) \( y = \frac{1}{1 + e^{-x}} \)

This is a sigmoid (S-shaped) function
Horizontal asymptote as \( x \to \infty \): \( y = 1 \)
Horizontal asymptote as \( x \to -\infty \): \( y = 0 \)
y-intercept at (0, 0.5)
No vertical asymptotes
Real World Connection: The sigmoid function (problem iv) is used in artificial intelligence, biology to model population growth, and even in psychology to describe learning curves!