Welcome to Sketching Curves!
In this chapter, we are going to learn how to move from looking at an equation to drawing a visual "story" of how that equation behaves. Sketching is different from drawing; we aren't worried about using a ruler for every millimeter. Instead, we want to capture the key features—the "personality" of the curve!
Being able to sketch curves is like having a superpower in Mathematics B (MEI). It helps you visualize solutions to equations and understand complex functions at a single glance. Don't worry if you find it a bit abstract at first; we will break it down step-by-step.
1. The "Anchor Points": Intersections
The first step in any sketch is finding where the curve crosses the axes. These are your "anchor points" that hold the sketch in place.
Crossing the y-axis
To find where the curve crosses the y-axis, we set \(x = 0\). This is usually the easiest point to find because you just plug zero into your formula.
Example: For \(y = x^2 + 5x + 6\), if \(x = 0\), then \(y = 6\). So, the curve crosses at (0, 6).
Crossing the x-axis
To find where the curve crosses the x-axis (the roots), we set \(y = 0\). This often requires you to solve an equation by factorising or using the quadratic formula.
Common Mistake: Students often forget that a curve might not cross the x-axis at all! If your quadratic equation has no real roots (the discriminant \(b^2 - 4ac < 0\)), the curve floats above or below the axis.
Key Takeaway:
Always find your intercepts first! Think of them as the "GPS coordinates" for your graph.
2. Mastering the Parabola (Quadratics)
Quadratic curves (parabolas) are symmetrical "U" or "n" shapes. We use the completing the square method to find their most important feature: the turning point.
The Magic Formula
If you write your equation in the form \(y = a(x + p)^2 + q\):
1. The turning point is at \((-p, q)\).
2. The line of symmetry is the vertical line \(x = -p\).
3. If \(a > 0\), it's a minimum (a happy "U" shape).
4. If \(a < 0\), it's a maximum (a sad "n" shape).
Memory Aid: "Positive is Happy, Negative is Sad." A positive \(x^2\) makes a smile; a negative \(x^2\) makes a frown!
Key Takeaway:
Completing the square tells you exactly where the curve "turns around." If you see \((x - 3)^2 + 2\), the curve turns at (3, 2). Notice how the sign of the number inside the bracket flips!
3. Polynomials and Repeated Roots
When sketching higher-degree polynomials like cubics (where the highest power is \(x^3\)), we look at the roots.
The "Trampoline" Rule
Sometimes a root is "repeated," like in \(y = (x - 2)^2(x + 1)\).
• A single root (like \(x = -1\) here) goes straight through the axis.
• A squared root (like \((x - 2)^2\)) just touches the axis and bounces back, like someone jumping on a trampoline. This is a stationary point.
Did you know? The "end behavior" of a cubic depends on the \(x^3\) term. If it's positive, the graph starts "down" on the left and ends "up" on the right. If it's negative, it starts "up" and ends "down."
Key Takeaway:
Look for brackets with powers. A power of 2 means the curve touches the axis at that point but doesn't cross it.
4. Reciprocal Graphs: \(y = \frac{a}{x}\) and \(y = \frac{a}{x^2}\)
These graphs are special because they have asymptotes. An asymptote is a line that the curve gets closer and closer to but never actually touches.
The Graph of \(y = \frac{a}{x}\) (The Hyperbola)
• If \(a\) is positive, the curve lives in the top-right and bottom-left quadrants.
• It has a vertical asymptote at \(x = 0\) (because you can't divide by zero!) and a horizontal asymptote at \(y = 0\).
The Graph of \(y = \frac{a}{x^2}\) (The Volcano)
• Because \(x^2\) is always positive, both "arms" of the graph are above the x-axis (if \(a > 0\)).
• This looks a bit like a volcano or a chimney. It also has asymptotes at \(x = 0\) and \(y = 0\).
Analogy: Asymptotes are like electric fences. The curve wants to get as close as possible to them, but if it touches them, the math "breaks" (like dividing by zero), so it stays infinitely close but separate.
Key Takeaway:
When sketching reciprocals, always draw your asymptotes as dashed lines first. They act as the "walls" that guide your curve.
5. Using Calculus in Sketching
To make your sketch really accurate, you can use differentiation to find stationary points. These are points where the gradient is zero (\(\frac{dy}{dx} = 0\)).
Types of Stationary Points:
1. Local Maximum: The top of a hill.
2. Local Minimum: The bottom of a valley.
3. Stationary Point of Inflection: A "shelf" where the curve flattens out momentarily before continuing in the same direction (like \(y = x^3\) at the origin).
Quick Review: To find these, find \(\frac{dy}{dx}\), set it to 0, and solve for \(x\). Then, plug \(x\) back into the original \(y\) equation to find the height!
Key Takeaway:
Stationary points are the "peaks and troughs" of your curve. Marking these (and labeling them as max, min, or inflection) makes your sketch look professional.
Final Checklist for a Perfect Sketch
Before you finish, check your drawing for these four things:
• Intersects: Are all axis crossings labeled with their values?
• Turning Points: Are the coordinates of peaks and valleys shown?
• Asymptotes: Are they drawn as dashed lines and labeled with their equations (e.g., \(x = 0\))?
• Shape: Does the overall "flow" match the type of equation (e.g., a cubic shouldn't look like a quadratic)?
Don't worry if your curves look a bit wobbly at first! The examiners are looking for the correct features and labels, not artistic perfection. Keep practicing, and soon you'll be able to "see" the curve just by looking at the algebra!