Accurate Graphing of Functions

Explanation

INTRODUCTION TO GRAPHING FUNCTIONS

Accurately graphing a function means not only sketching its approximate shape but also analyzing the function's most important features. These include zeros (x-intercepts), stationary points, intervals of increase and decrease, possible extrema, asymptotes, concavity, and inflection points. Each of these features contributes to the correct shape of the graph.

STEPS FOR GRAPHING A FUNCTION

To graph a function correctly, a specific sequence of steps is followed:

DETERMINE THE DOMAIN OF THE FUNCTION: Check for which values of x the function is defined (e.g., division by zero is not allowed, the square root of a negative number is not defined in the real number system, etc.).
FIND THE ZEROS (X-INTERCEPTS) OF THE FUNCTION: Solve the equation f(x) = 0. These values represent the points where the graph intersects the x-axis.
FIND THE FIRST DERIVATIVE OF THE FUNCTION AND SOLVE f'(x) = 0: These are candidates for local maxima, minima, or stationary inflection points.
ANALYZE INCREASING AND DECREASING INTERVALS: Determine where the function is increasing (f'(x) > 0) and where it is decreasing (f'(x) < 0) based on the sign of the first derivative.
CHECK FOR LOCAL EXTREMA: By observing the change in the sign of the derivative around stationary points, we can determine if it is a maximum, a minimum, or just a flat point (saddle/stationary inflection point).
DETERMINE CONCAVITY AND INFLECTION POINTS: Calculate the second derivative, f''(x). If f''(x) > 0, the graph is concave upwards (convex). If f''(x) < 0, it is concave downwards. Points where the concavity changes are inflection points (typically where f''(x) = 0 or is undefined, and changes sign).
CHECK FOR ASYMPTOTES (IF ANY): If the function contains fractions or logarithmic expressions, check the behavior for very large or very small values of x, and where denominators are zero.
CALCULATE THE FUNCTION'S VALUES FOR SOME X-VALUES: Substitute specific values of x to obtain points through which the graph passes (e.g., x = 0, x = 1, x = -1 ...).

IMPORTANT ELEMENTS OF THE GRAPH

Y-INTERCEPT: Determined by calculating f(0).
X-INTERCEPTS (ZEROS): Determined from the equation f(x) = 0.
STATIONARY POINTS AND EXTREMA: Found using the first derivative.
CONCAVITY AND INFLECTION POINTS: Found using the second derivative.
ASYMPTOTES: Horizontal, vertical, or slant (if the function has them).
INCREASING/DECREASING INTERVALS: Determined by the sign of the first derivative.

EXAMPLE

Let the function be f(x) = x³ - 3x.

DOMAIN: The entire set of real numbers.
ZEROS: Solve x³ - 3x = 0 → x(x² - 3) = 0 → x(x - √3)(x + √3) = 0. Zeros are x = 0, x = √3, x = -√3.
FIRST DERIVATIVE: f'(x) = 3x² - 3. Solve f'(x) = 0 → 3(x² - 1) = 0 → x = ±1. These are stationary points.
SIGN OF THE FIRST DERIVATIVE:
- For x < -1 (e.g., x = -2), f'(-2) = 3(-2)² - 3 = 12 - 3 = 9 > 0. The function is increasing.
- For -1 < x < 1 (e.g., x = 0), f'(0) = 3(0)² - 3 = -3 < 0. The function is decreasing.
- For x > 1 (e.g., x = 2), f'(2) = 3(2)² - 3 = 12 - 3 = 9 > 0. The function is increasing.
LOCAL EXTREMA:
- At x = -1, the function changes from increasing to decreasing, so there is a local maximum. f(-1) = (-1)³ - 3(-1) = -1 + 3 = 2. Point: (-1, 2).
- At x = 1, the function changes from decreasing to increasing, so there is a local minimum. f(1) = (1)³ - 3(1) = 1 - 3 = -2. Point: (1, -2).
SECOND DERIVATIVE: f''(x) = 6x. Set f''(x) = 0 → 6x = 0 → x = 0.
- For x < 0, f''(x) < 0 (concave downwards).
- For x > 0, f''(x) > 0 (concave upwards).
  The point (0, f(0)) = (0, 0) is an inflection point where concavity changes.
The function has no asymptotes.
OTHER POINTS: We already have f(0) = 0. Let's check f(2) = 2³ - 3(2) = 8 - 6 = 2. Point (2, 2).

CONCLUSION

Accurately graphing functions is based on an analytical investigation of their properties. By examining domains, derivatives, changes in direction, concavity, and other key features in the correct order, a complete picture of the function's behavior across its entire domain can be created. This is not just a drawing exercise but a structured approach to a deeper understanding of functions.