More about Graphs of Functions
Master absolute value transformations, find vertical and horizontal asymptotes instantly, and learn how to solve complex equations and inequalities graphically without touching algebra.
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The Core Concept
The Geometry of Equations: Algebraic equations are just numerical descriptions of geometric collisions. When we solve
f(x) = g(x), we are geometrically finding where the curve y = f(x) and the line y = g(x) collide. The x-coordinates of these intersection points are our real roots.DSE More about Graphs of Functions Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Graphical Solutions | To solve f(x) = g(x) graphically:1. Plot y = f(x) and y = g(x) on the same axes.2. Find where the curves intersect. | The x-coordinates of the intersection points are the real roots of the equation. |
| 2. Graphical Inequalities | To solve f(x) > g(x) graphically:Look for the intervals of x where the curve y = f(x) lies above y = g(x). | If the inequality is f(x) < g(x), look for the intervals where y = f(x) lies below y = g(x). |
| 3. Absolute Value: y = |f(x)| | Take the graph of y = f(x). Any portion below the x-axis (y < 0) is reflected upward to become positive. The parts already above the x-axis stay unchanged. | Think "Fold Bottom Up": bottom curves reflect to the top, top curves remain untouched. |
| 4. Absolute Value: y = f(|x|) | Take the graph of y = f(x) for x ≥ 0 only (right side). Erase the entire left side (x < 0), then mirror the right side to the left. | Think "Copy Right to Left": erase the left side entirely, then duplicate the right side onto the left like a mirror. |
| 5. Reciprocal Asymptotes | For a rational function like y = 1/(x − h) + k:• Vertical Asymptote (VA): x = h• Horizontal Asymptote (HA): y = k | • VA occurs where the denominator equals 0. • HA occurs as x approaches infinity ( x → ±∞).For y = (ax + b)/(cx + d), HA = a/c (ratio of x coefficients). |
💡 Exam Traps & Secrets
⚠️ The "Which Way to Fold?" Absolute Value TrapStudents constantly mix up the reflection rules for
For
For
y = |f(x)| and y = f(|x|).For
y = |f(x)|: Think "Fold Bottom Up" (bottom curves reflect to the top, top curves remain untouched).For
y = f(|x|): Think "Copy Right to Left" (erase the left side entirely, then duplicate the right side onto the left like a mirror).⚠️ The "Real Roots" Intersection Trick (Paper 2)In Paper 2, you are often asked to find the number of real roots of an equation. Do not waste time using quadratic formulas or long division. Just sketch the left side as
y = f(x) and the right side as y = g(x) and count the number of times the two graphs intersect.⚠️ The Rational Function HA ShortcutIf you are given a rational function like
y = (ax + b)/(cx + d), you can find the Horizontal Asymptote instantly by taking the ratio of the x coefficients: HA: y = a/c. Example: For y = (3x − 5)/(2x + 1), the horizontal asymptote is simply y = 3/2.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Visualizing the intersection of y = |x − 2| and y = 3)Solve the absolute value equation
|x − 2| = 3 graphically.(Identifying asymptotes visually)Sketch the rational function
y = 1/(x − 2) + 3 and state its horizontal and vertical asymptotes.(Graphical inequalities)Use the graphs of
y = x² and y = 2x + 3 to find the range of x that satisfies the inequality x² ≤ 2x + 3 graphically.More about Graphs of Functions Exercises: DSE-Style Practice
DSE-style practice sets from the SketchMath question bank, with solutions for every MC and long question.
Functions and Graphs · Form 4
34 MC questions · 4 long questions
Frequently Asked Questions
Plot both sides as separate functions and find the x-coordinates of their intersection points. This is especially useful for absolute value equations, logarithmic equations, and any equation where algebraic manipulation is messy or time-consuming.
For y = |f(x)|: any part of the graph below the x-axis gets reflected upward ("Fold Bottom Up"). For y = f(|x|): erase the left side (x < 0), then duplicate the right side (x ≥ 0) to the left ("Copy Right to Left"). These are the two most confused transformations in Paper 2.
Vertical asymptote: set the denominator = 0 and solve. Horizontal asymptote: for y = (ax + b)/(cx + d), HA = a/c (ratio of x coefficients). For y = 1/(x − h) + k, VA is x = h and HA is y = k. These shortcuts save minutes on Paper 2 MC questions.