How to Sketch a Curve

The term 'curve sketching' refers to the process of drawing a curve with sufficient detail to convey its overall shape and position. To sketch the curve of a function f(x) on a Cartesian plane, precise values of f(x) are not necessary, but the following information is essential:

1. Behavior at Infinity: What does the curve behave like as x approaches large values (both positive and negative)? This is determined by analyzing the equation of the curve.
2. Discontinuities: Are there any breaks in the curve? If so, where are they and what are their characteristics? Again, this is determined by examining the equation of the curve.
3. Turning Points: Are there any points where the curve changes direction? We typically find these using differential calculus.
4. X-axis Intersections: Where, if at all, does the curve intersect the x-axis? This is usually identified through algebraic methods to solve f(x) = 0.

An Example: Analyzing a Cubic Equation

Consider the following exam question from a 1990 University of Cambridge entrance paper. Notably, part (vii) is likely the simplest because it deals with the well-known curve of y = x³. However, examining the seven different parameter choices for b and c reveals deeper insights.

Before delving into each case, let's revisit the four questions mentioned previously and explore some general answers:

1. Behavior as x Approaches Positive and Negative Infinity: Given that this is a cubic polynomial with a positive leading coefficient, we can conclude that y will increase positively as x grows positively, and y will decrease negatively as x becomes negative.
2. Discontinuities: Since this is a polynomial function, it is continuous for all real x, meaning there are no discontinuities.
3. Turning Points: Turning points occur when the first derivative equals zero. If the first derivative is zero and the second derivative is positive, it's a minimum turning point. Conversely, if the first derivative is zero and the second derivative is negative, it indicates a maximum turning point. If both the first and second derivatives are zero, and the third derivative is positive, we have a point of inflection.

The first video, "Curve Sketching - First & Second Derivatives - Graphing Rational Functions & Asymptotes - Calculus," provides a visual guide to understanding these concepts in depth.

Identifying Roots

To find the roots of our equation, which indicate where the curve intersects or touches the x-axis, we can factorize and apply the quadratic formula. First, we note that y = x(x² - 2bx + c²), meaning x = 0 is always a root. The other two roots, if they exist, can be determined by x = b ± √(b² - c²). Understanding the conditions for b and c from parts (i) to (vii) can provide further clarity.

Sketching Our Curves

Now, let’s begin sketching.

Part (i) states that 2b < √3c, which implies that 4b² - 3c² < 0. Referring to our earlier equation for the first derivative dy/dx, we can conclude that there are no real roots, thus no turning points exist. Additionally, we find that b < √3c/2 < c, leading to b² - c² < 0. As a result, our curve only intersects the x-axis at x = 0. We now have enough information to sketch the curve.

Part (ii) indicates that b < c, confirming that the curve intersects the x-axis only at x = 0. With 2b = √3c, we identify a single root for the first derivative at x = 2b/3. Evaluating the second derivative at this point reveals it is zero, and the third derivative is positive, indicating a point of inflection at x = 2b/3. Here's a sketch:

The second video, "A summary of curve sketching - Calc 1," offers a concise overview of the curve sketching process, reinforcing the concepts discussed.

Continuing with Parts (iii) to (vii), we can derive similar insights regarding x-axis intersections, turning points, and their implications, culminating in the final sketches.

Checking My Sketches Using ggplot2 in R

To verify the accuracy of my sketches, I utilized the powerful ggplot2 package in R. Below is a function I created to visualize the curves based on parameters b and c:

library(ggplot2)

plot_graph <- function(b, c, part) {

ggplot() +

xlim(-10, 10) +

geom_vline(xintercept = 0) +

geom_hline(yintercept = 0) +

geom_function(fun = function(x) {x^3 - 2*b*x^2 + c^2*x}, color = "blue") +

theme_minimal() +

labs(title = paste("Part", part))

}

Using this function, I generated plots that correspond to the conditions outlined in parts (i) to (vii) and compiled them into a single image.

library(patchwork)

p1 <- plot_graph(b = 2, c = 6, part = "(i)")

p2 <- plot_graph(b = 4, c = 8/sqrt(3), part = "(ii)")

p3 <- plot_graph(b = 8, c = 8.5, part = "(iii)")

p4 <- plot_graph(b = 8, c = 8, part = "(iv)")

p5 <- plot_graph(b = 6, c = 5.5, part = "(v)")

p6 <- plot_graph(b = 4, c = 0, part = "(vi)")

p7 <- plot_graph(b = 0, c = 0, part = "(vii)")

(p1 + p2) / (p3 + p4) / (p5 + p6 + p7)

The result demonstrates a commendable effort, though I may have slightly overestimated the spacing between maxima and minima in my sketches. This leads me to conclude that while I possess mathematical skill, my artistic abilities may need improvement.

Do you frequently engage in curve sketching? I welcome your thoughts on my approach—please feel free to share your feedback.