Graph-based understanding of functions

This chapter establishes a foundational understanding of functions through their graphical representations. It explores key concepts such as domain restrictions, graph transformations, modulus functions, and piecewise definitions, all critical for visualising functional behavior. Mastery of these topics is essential for interpreting complex functions and solving analytical problems frequently encountered in examinations.

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Chapter Contents

|

| Topic |

|---|-------| | 1 | Graph of a function | | 2 | Domain restrictions | | 3 | Graph transformations | | 4 | Modulus functions | | 5 | Piecewise functions |

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We begin with Graph of a function.

Part 1: Graph of a function

Graph of a Function

Overview

The graph of a function is the visual language of algebra. In CMI-style problems, graphs are not only about drawing curves; they are used to understand domain, range, monotonicity, symmetry, number of solutions, transformations, and composition. A strong graph sense turns many algebraic questions into simple geometric observations. ---

Learning Objectives

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What a Graph Represents

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Vertical Line Test

Examples

• $y=x^2$ passes the test

• $y=\sqrt{x}$ passes the test
  • $x^2+y^2=1$ does not pass the test as a whole curve
  • a circle is not the graph of a function of $x$
  ⚠️ Common Confusion

  A relation may fail to be a function even if it looks smooth.
  
  For example, the circle
  $\qquad x^2+y^2=1$
  gives two values of $y$ for most $x$ between $-1$ and $1$.

  ---

  Domain and Range from the Graph

  📖 Domain and Range

  • Domain = all $x$-values for which the graph has points

  • Range = all $y$-values attained by the graph
  💡 How to Read Them

  • To find the domain, project the graph onto the $x$-axis.

  • To find the range, project the graph onto the $y$-axis.
  Examples
  • For $y=x^2$, domain is all real numbers, range is $y\ge 0$
  • For $y=\sqrt{x}$, domain is $x\ge 0$, range is $y\ge 0$
    • For $y=\dfrac{1}{x}$, domain is $x\ne 0$, range is $y\ne 0$
    ---

    Intercepts and Zeros

    📐 Intercepts

    • $x$-intercepts occur where $y=0$, so they solve

    $\qquad f(x)=0$

    • $y$-intercept occurs where $x=0$, so its value is

    $\qquad f(0)$ if defined
    ❗ Graph Interpretation of Equations

    • Number of solutions of $f(x)=0$ = number of intersections with the $x$-axis

    • Number of solutions of $f(x)=k$ = number of intersections with the horizontal line $y=k$

    • Number of solutions of $f(x)=g(x)$ = number of intersections of the two graphs
    ---

    Increasing and Decreasing

    📖 Monotonic Behavior

    A function is:
    
    

    • increasing on an interval if larger $x$ gives larger $f(x)$

    
    

    • decreasing on an interval if larger $x$ gives smaller $f(x)$

    💡 Visual Test

    Move from left to right:
    

    • if the graph goes upward, the function is increasing

    
    

    • if the graph goes downward, the function is decreasing

    This is one of the fastest graphical tools for inequalities and inverse functions. ---

    Maximum, Minimum, and Turning Points

    📖 Extrema

    A point where the function attains a highest local value is a local maximum.
    
    A point where the function attains a lowest local value is a local minimum.
    
    Together, these are called turning points or extrema.

    ❗ Graph Reading

    • Highest point on the whole graph section $\rightarrow$ global maximum

    • Lowest point on the whole graph section $\rightarrow$ global minimum

    • A parabola $y=x^2$ has a minimum at $(0,0)$
    ---

    Sign of a Function

    📐 Positive and Negative Regions

    • $f(x)>0$ where the graph lies above the $x$-axis

    • $f(x)<0$ where the graph lies below the $x$-axis

    • $f(x)=0$ where the graph meets the $x$-axis
    This is extremely useful in inequality problems. ---

    Symmetry

    📐 Even and Odd Graphs

    A function is even if
    $\qquad f(-x)=f(x)$
    
    Its graph is symmetric about the $y$-axis.
    
    A function is odd if
    $\qquad f(-x)=-f(x)$
    
    Its graph is symmetric about the origin.

    Examples
    • $f(x)=x^2$ is even
    • $f(x)=x^3$ is odd
    • $f(x)=|x|$ is even
    💡 Quick Graph Test

    • Mirror symmetry across the $y$-axis $\rightarrow$ even

    • 180-degree rotational symmetry about the origin $\rightarrow$ odd
    ---

    Standard Parent Graphs

    📐 Graphs You Must Know Instantly

    • Constant function:

    
    $\qquad y=c$
    horizontal line
    
    

    • Identity function:

    
    $\qquad y=x$
    straight line through origin, slope $1$
    
    

    • Quadratic:

    
    $\qquad y=x^2$
    upward parabola
    
    

    • Cubic:

    
    $\qquad y=x^3$
    S-shaped, odd symmetry
    
    

    • Absolute value:

    
    $\qquad y=|x|$
    V-shaped graph
    
    

    • Reciprocal:

    
    $\qquad y=\dfrac{1}{x}$
    two branches, axes are asymptotes
    
    

    • Square root:

    
    $\qquad y=\sqrt{x}$
    starts at $(0,0)$ and moves rightward
    
    

    • Modulus-shifted forms:

    
    $\qquad y=|x-a|+b$
    V-shape shifted to vertex $(a,b)$

    ---

    Transformations of Graphs

    📐 Basic Transformations

    If the graph of $y=f(x)$ is known, then:
    
    

    • $y=f(x)+k$

    
    shifts graph up by $k$
    
    

    • $y=f(x)-k$

    
    shifts graph down by $k$
    
    

    • $y=f(x-h)$

    
    shifts graph right by $h$
    
    

    • $y=f(x+h)$

    
    shifts graph left by $h$
    
    

    • $y=af(x)$

    
    vertical stretch/compression by factor $|a|$
    
    

    • $y=f(ax)$

    
    horizontal compression/stretch
    
    

    • $y=-f(x)$

    
    reflection in the $x$-axis
    
    

    • $y=f(-x)$

    
    reflection in the $y$-axis

    ⚠️ Very Common Trap

    Inside changes and outside changes behave differently:
    
    

    • $f(x-2)$ shifts right

    
    

    • $f(x)+2$ shifts up

    
    
    The signs are not interpreted in the same way.

    ---

    Piecewise Graphs

    📖 Piecewise Functions

    A piecewise function uses different formulas on different intervals.
    
    Example:
    $\qquad f(x)=\begin{cases}<br>x+1, & x<0 \\ <br>x^2, & x\ge 0 <br>\end{cases}$
    
    To graph such a function:
    

    • graph each rule only on its own interval,

    
    

    • mark open or closed endpoints carefully.

    ❗ Endpoint Convention

    • open circle $\rightarrow$ endpoint not included

    • filled circle $\rightarrow$ endpoint included
    ---

    One-One Functions and Horizontal Line Test

    📐 Horizontal Line Test

    A function is one-one if every horizontal line intersects its graph at at most one point.

    This matters because:
    • one-one functions have inverses that are also functions
    • the graph of $f^{-1}$ is the reflection of the graph of $f$ in the line
    $\qquad y=x$ ::: Examples
    • $y=x^3$ is one-one
    • $y=x^2$ is not one-one on all real numbers
    • $y=x^2$ becomes one-one if domain is restricted to $x\ge 0$
    ---

    Graphs and Number of Solutions

    ❗ Very Useful Exam View

    To solve equations graphically:
    
    

    • $f(x)=0$

    
    count intersections with the $x$-axis
    
    

    • $f(x)=k$

    
    count intersections with the line $y=k$
    
    

    • $f(x)=g(x)$

    
    count intersections of the two graphs
    
    

    • $f(x)\ge 0$

    
    find where graph is on or above the $x$-axis
    
    

    • $f(x)\le g(x)$

    
    compare which graph lies above the other

    ---

    Minimal Worked Examples

    Example 1 For $f(x)=|x-2|+1$, the graph is obtained from $y=|x|$ by shifting:
    • right by $2$
    • up by $1$
    So the vertex is at $\qquad (2,1)$ --- Example 2 For $f(x)=\dfrac{1}{x-3}$, the graph of $y=\dfrac{1}{x}$ is shifted right by $3$. So:
    • vertical asymptote is
    $\qquad x=3$
    • horizontal asymptote remains
    $\qquad y=0$ ---

    CMI Strategy

    💡 How to Read a Function Graph Quickly

    • Check whether it is really a function using the vertical line test.

    
    

    • Read domain and range before doing algebra.

    
    

    • Mark intercepts and special points.

    
    

    • Notice symmetry, monotonicity, and turning points.

    
    

    • For equations, think intersections.

    
    

    • For inequalities, think above/below the relevant line.

    
    

    • For transformed graphs, identify the parent function first.

    ---

    Common Mistakes

    ⚠️ Avoid These Errors

    • ❌ Confusing relation with function

    ✅ Use the vertical line test

    • ❌ Forgetting that domain comes from allowed $x$-values

    • ❌ Mixing up horizontal and vertical shifts

    • ❌ Saying $f(x)>0$ where the graph is to the right of the $y$-axis

    ✅ Positivity means above the $x$-axis

    • ❌ Ignoring open and closed endpoints in piecewise graphs

    • ❌ Thinking every function has an inverse function on its full domain
    ---

    Practice Questions

    :::question type="MCQ" question="Which of the following curves is the graph of a function of $x$?" options=["$x^2+y^2=1$","$y=|x|$","$x=3$","$y^2=x$"] answer="B" hint="Use the vertical line test." solution="We test each option.
  • $x^2+y^2=1$ is a circle, so a vertical line usually cuts it twice.
  • $y=|x|$ gives exactly one value of $y$ for each real $x$.
  • $x=3$ is a vertical line, so it fails the vertical line test.
  • $y^2=x$ gives two values of $y$ for each $x>0$.
  Hence only $y=|x|$ is the graph of a function of $x$. Therefore the correct option is $\boxed{B}$." ::: :::question type="NAT" question="The graph of $y=(x-2)^2+3$ has vertex at $(a,b)$. Find $a+b$." answer="5" hint="Compare with the shifted parabola form $y=(x-h)^2+k$." solution="The graph $\qquad y=(x-2)^2+3$ is obtained from $\qquad y=x^2$ by shifting right by $2$ and up by $3$. So the vertex is $\qquad (a,b)=(2,3)$ Hence $\qquad a+b=2+3=5$ Therefore the answer is $\boxed{5}$." ::: :::question type="MSQ" question="Which of the following statements are true?" options=["For a function $f$, solutions of $f(x)=0$ are the $x$-intercepts of its graph","If a graph is symmetric about the $y$-axis, the function is even","If a horizontal line cuts the graph more than once, then the graph is not a function","For $f(x)=\dfrac1x$, the domain is $x\ne 0$"] answer="A,B,D" hint="Separate vertical line test from horizontal line test." solution="1. True. $f(x)=0$ means $y=0$, so these are exactly the $x$-intercepts.
  • True. Symmetry about the $y$-axis corresponds to $f(-x)=f(x)$.
  • False. A graph may still be a function even if a horizontal line cuts it more than once. That only means it is not one-one.
  • True. For $f(x)=\dfrac1x$, $x=0$ is not allowed.
  Hence the correct answer is $\boxed{A,B,D}$." ::: :::question type="SUB" question="Explain graphically why the equation $x^2=2-x$ has exactly two real solutions." answer="The parabola $y=x^2$ and the line $y=2-x$ intersect in exactly two points." hint="Interpret the equation as intersection of two graphs." solution="Consider the two graphs $\qquad y=x^2$ and $\qquad y=2-x$ The first is an upward-opening parabola, and the second is a straight line with negative slope. The solutions of $\qquad x^2=2-x$ are exactly the $x$-coordinates of the intersection points of these two graphs. Now the line cuts the parabola in exactly two distinct points. Hence the equation has exactly two real solutions. So graphically, the reason is that the two graphs intersect exactly twice." ::: ---

  Summary

  ❗ Key Takeaways for CMI

  • The graph of $f$ is the set of points $(x,f(x))$.

  
  

  • The vertical line test decides whether a curve is a function of $x$.

  
  

  • Domain and range are read from the horizontal and vertical spread of the graph.

  
  

  • Zeros, signs, maxima, minima, and monotonicity are all visible graphically.

  
  

  • Graph transformations are one of the fastest ways to understand new functions.

  
  

  • Equations and inequalities often become intersection and position questions on graphs.

  ---

  💡 Next Up

  Proceeding to Domain restrictions.

  ---

  Part 2: Domain restrictions

  Domain restrictions

  Overview

  Domain restrictions tell us which inputs are allowed for a function. In school algebra this begins with avoiding division by zero and square roots of negative numbers. In CMI-style questions, domain analysis is deeper: it helps detect hidden restrictions, reject fake solutions, and understand where a graph really exists. A correct function study starts with the domain. ---

  Learning Objectives

  ❗ By the End of This Topic

  After studying this topic, you will be able to:
  

  • Find the domain of algebraic expressions involving fractions, roots, and logarithms.

  
  

  • Detect hidden restrictions created by simplification.

  
  

  • Use graph-based reasoning to see where a function is defined or undefined.

  
  

  • Distinguish between algebraic form and actual domain.

  
  

  • Reject extraneous solutions in equations and inequalities by checking validity.

  ---

  Core Idea

  📖 Domain of a function

  The domain of a function is the set of all input values for which the function is defined.
  
  If a formula contains an operation that is not valid for some values, those values must be removed from the domain.

  ❗ Main Sources of Restrictions

  In real-variable algebra, restrictions usually come from:
  

  • denominators

  
  

  • even roots

  
  

  • logarithms

  
  

  • inverse trigonometric or other special-function conditions

  
  

  • piecewise definitions

  
  

  • hidden expressions after substitution

  ---

  Standard Domain Rules

  📐 Basic Rules in the Real Number System

  For real-valued functions:
  
  

  • $\dfrac{1}{g(x)}$ is defined only when

  
  $\qquad g(x) \ne 0$
  
  

  • $\sqrt{g(x)}$ is defined only when
    $\qquad g(x) \ge 0$
    
    
    • $\sqrt[n]{g(x)}$ with even $n$ requires
      $\qquad g(x) \ge 0$
      
      
      • $\sqrt[n]{g(x)}$ with odd $n$ is defined for every real $g(x)$
        
        
        • $\log_a(g(x))$ is defined only when
        
        $\qquad g(x) > 0$
        and also the base satisfies
        $\qquad a>0,\ a\ne 1$
        
        
        • If several restrictions occur together, all must hold simultaneously.

  ---

  Denominator Restrictions

  📐 Division by zero is never allowed

  If
  $\qquad f(x)=\dfrac{P(x)}{Q(x)}$
  
  then the domain is all real numbers except the zeros of $Q(x)$.
  
  Examples:
  

  • $\dfrac{1}{x-3}$ has domain $\mathbb{R}\setminus\{3\}$

  
  

  • $\dfrac{x+1}{x^2-4}$ has domain $\mathbb{R}\setminus\{-2,2\}$

  ⚠️ Hidden Cancellation Trap

  The expression
  $\qquad \dfrac{x^2-1}{x-1}$
  
  simplifies algebraically to
  $\qquad x+1$
  
  but the original expression is still undefined at
  $\qquad x=1$
  
  So its domain is
  $\qquad \mathbb{R}\setminus\{1\}$
  
  not all real numbers.

  ---

  Even Root Restrictions

  📐 Square roots and even roots

  For real-valued functions:
  

  • $\sqrt{x}$ requires $x\ge 0$
    
    • $\sqrt{x-5}$ requires $x\ge 5$
      
      • $\sqrt{(x-1)(x-4)}$ requires
        $\qquad (x-1)(x-4)\ge 0$
        
        So domain questions often reduce to solving an inequality.

  💡 Interval Method Link

  Whenever the argument of an even root is a polynomial or rational expression, use sign analysis or interval method to find where it is non-negative.

  ---

  Logarithmic Restrictions

  📐 Logarithm domain rule

  For
  $\qquad \log_a(g(x))$
  
  we must have
  $\qquad g(x)>0$
  
  Examples:
  

  • $\log_2(x-3)$ requires $x>3$

  
  

  • $\log_5(x^2-9)$ requires $x^2-9>0$

  
  

  • $\log(x)+\log(1-x)$ requires

  
  $\qquad x>0$ and $1-x>0$

  ⚠️ Log Trap

  For logarithms, the argument must be strictly positive, not merely non-negative.
  
  So:
  

  • $\log(x)$ does not allow $x=0$

  
  

  • $\log(x^2)$ does not allow $x=0$

  ---

  Combined Restrictions

  📐 When multiple conditions appear

  To find the domain of a complicated expression, combine all conditions.
  
  Example:
  $\qquad f(x)=\dfrac{\sqrt{x-1}}{x^2-9}$
  
  We need:
  

  • $x-1\ge 0 \Rightarrow x\ge 1$

  
  

  • $x^2-9\ne 0 \Rightarrow x\ne 3,-3$

  
  
  Since $x\ge 1$, only $x=3$ matters from the denominator.
  
  Hence the domain is
  $\qquad [1,\infty)\setminus\{3\}$

  ---

  Graph-Based Understanding

  📖 What a graph tells us about domain

  From the graph of a function, the domain is the set of all $x$-coordinates for which the graph has at least one point.
  
  So:
  

  • if there is a vertical gap at some $x$, that value is not in the domain

  
  

  • if there is a hole, that input is excluded

  
  

  • if the graph starts from a point and continues rightward, the domain may be a half-line

  
  

  • if the graph appears only on separated pieces, the domain is a union of intervals

  💡 How to read domain from a graph

  To decide whether an $x$ is in the domain, imagine drawing a vertical line at that $x$:
  

  • if the graph contains at least one plotted point on that line, the input is allowed

  
  

  • if no point exists there, the input is not in the domain

  ---

  Holes, Breaks, and Endpoints

  📐 Graph patterns

  • Open circle at an endpoint means that point is not included.

  
  

  • Closed dot means the point is included.

  
  

  • Vertical asymptote means the function is not defined there.

  
  

  • Hole often appears after algebraic cancellation and indicates a removed domain point.

  Example If $\qquad f(x)=\dfrac{x^2-1}{x-1}$ then for $x\ne 1$, $\qquad f(x)=x+1$ So the graph is the line $y=x+1$ with a hole at $\qquad (1,2)$ This is a standard graph-based domain example. ---

  Minimal Worked Examples

  Example 1 Find the domain of $\qquad f(x)=\dfrac{1}{x^2-5x+6}$ Factor the denominator: $\qquad x^2-5x+6=(x-2)(x-3)$ So we need $\qquad (x-2)(x-3)\ne 0$ Hence $\qquad x\ne 2,\ 3$ Therefore the domain is $\qquad \boxed{\mathbb{R}\setminus\{2,3\}}$ --- Example 2 Find the domain of $\qquad f(x)=\sqrt{x^2-5x+6}$ We need $\qquad x^2-5x+6\ge 0$ Factor: $\qquad (x-2)(x-3)\ge 0$ By interval method, this holds for $\qquad x\le 2 \quad \text{or} \quad x\ge 3$ So the domain is $\qquad \boxed{(-\infty,2]\cup[3,\infty)}$ ---

  Extraneous Solutions and Domain Checking

  ❗ Why domain matters in equations

  Some equations produce answers that are algebraically possible but not allowed in the original expression.
  
  Example:
  If we solve
  $\qquad \sqrt{x-1}=x-3$
  
  we must still check:
  

  • $x-1\ge 0$

  
  

  • right side must be non-negative if both sides are equal to a square root

  
  
  So domain and sign checks are part of the solution.

  ---

  Common Errors

  ⚠️ Avoid These Errors
  • ❌ cancelling a factor and forgetting the removed value
  • ❌ allowing denominator zero
  • ❌ using $\ge 0$ for log arguments instead of $>0$
  • ❌ forgetting that even roots need non-negative input
  • ❌ solving equations without checking whether the answer lies in the original domain
  • ❌ reading a graph as continuous when there is actually a hole or open circle
  ✅ Correct habits:
  • write all restrictions first
  • combine them carefully
  • keep the original expression in mind
  • verify final answers in the original formula
  ---

  CMI Strategy

  💡 How to Approach Domain Questions

  • Scan the formula for denominators, roots, and logs.

  
  

  • Translate each into an inequality or exclusion.

  
  

  • Solve all restrictions together.

  
  

  • For graphical questions, read the set of allowed $x$-coordinates.

  
  

  • If the expression simplifies, keep the original restrictions.

  
  

  • In equation problems, check every obtained solution against the original domain.

  ---

  Practice Questions

  :::question type="MCQ" question="The domain of $\dfrac{1}{x^2-4}$ is" options=["$\mathbb{R}\setminus\\{-2,2\\}$","$\mathbb{R}\setminus\\{2\\}$","$(-2,2)$","$\mathbb{R}$"] answer="A" hint="The denominator must be nonzero." solution="We need $\qquad x^2-4\ne 0$ Factor: $\qquad x^2-4=(x-2)(x+2)$ So the forbidden values are $\qquad x=2,\ -2$ Hence the domain is $\qquad \boxed{\mathbb{R}\setminus\{-2,2\}}$ Therefore the correct option is $\boxed{A}$." ::: :::question type="NAT" question="How many integers belong to the domain of $\sqrt{(x-1)(x-4)}$ in the interval $[-2,6]$?" answer="6" hint="The square root requires $(x-1)(x-4)\ge 0$." solution="We need $\qquad (x-1)(x-4)\ge 0$ This holds for $\qquad x\le 1 \quad \text{or} \quad x\ge 4$ Now restrict to the interval $[-2,6]$. The integers satisfying this are $\qquad -2,-1,0,1,4,5,6$ There are $7$ such integers. Hence the answer is $\boxed{7}$." ::: :::question type="MSQ" question="Which of the following statements are true?" options=["$\log(x^2)$ is defined for all real $x$ except $0$","$\sqrt{x^2}$ is defined for all real $x$","$\dfrac{x^2-1}{x-1}$ is defined for all real $x$ except $1$","$\log(1-x)$ is defined for all $x\le 1$"] answer="A,B,C" hint="Check each definition carefully." solution="1. True. Since $x^2>0$ for all $x\ne 0$, $\log(x^2)$ is defined for all real $x$ except $0$.
  • True. Since $x^2\ge 0$ for every real $x$, $\sqrt{x^2}$ is defined for all real $x$.
  • True. Although $\dfrac{x^2-1}{x-1}$ simplifies to $x+1$, the original expression is undefined at $x=1$.
  • False. For $\log(1-x)$ we need
  $\qquad 1-x>0 \Rightarrow x<1$ So $x=1$ is not allowed. Hence the correct answer is $\boxed{A,B,C}$." ::: :::question type="SUB" question="Find the domain of $f(x)=\dfrac{\sqrt{x-2}}{x^2-5x+6}$." answer="$[2,\infty)\setminus\\{3\\}$" hint="Combine the square-root restriction with the denominator restriction." solution="We need two conditions. First, from the square root: $\qquad x-2\ge 0 \Rightarrow x\ge 2$ Second, from the denominator: $\qquad x^2-5x+6\ne 0$ Factor: $\qquad x^2-5x+6=(x-2)(x-3)$ So we must exclude $\qquad x=2,\ 3$ Combining with $x\ge 2$, the domain is $\qquad (2,\infty)\setminus\{3\}$ Therefore the domain is $\boxed{(2,\infty)\setminus\{3\}}$." ::: ---

  Summary

  ❗ Key Takeaways for CMI

  • The domain is the set of all valid inputs.

  
  

  • Denominators must be nonzero.

  
  

  • Even roots require non-negative inputs.

  
  

  • Logarithm arguments must be strictly positive.

  
  

  • Simplifying an expression does not remove original restrictions.

  
  

  • Graphs reveal domain through allowed $x$-coordinates, holes, asymptotes, and endpoints.

  ---

  💡 Next Up

  Proceeding to Graph transformations.

  ---

  Part 3: Graph transformations

  Graph Transformations

  Overview

  Graph transformations help us understand how the graph of a function changes when the formula is modified. Instead of plotting from scratch every time, we learn how shifts, stretches, compressions, reflections, and modulus operations affect an already known graph. In CMI-style problems, the real test is careful interpretation: whether a change acts inside the input or outside the function value. ---

  Learning Objectives

  ❗ By the End of This Topic

  After studying this topic, you will be able to:
  

  • identify horizontal and vertical shifts correctly

  
  

  • distinguish transformations acting on $x$ from those acting on $f(x)$

  
  

  • understand vertical and horizontal scaling

  
  

  • handle reflections in the $x$-axis and $y$-axis

  
  

  • interpret graphs of $|f(x)|$ and $f(|x|)$

  
  

  • build transformed graphs quickly from a base graph

  ---

  Base Idea

  📖 Start with a Known Graph

  Suppose the graph of
  
  $\qquad y=f(x)$
  
  is known.
  
  A transformed equation such as
  
  $\qquad y=f(x-a)+b$
  
  or
  
  $\qquad y=af(bx)$
  
  can often be understood directly from the graph of $y=f(x)$ without recomputing many points.

  ❗ Golden Rule

  Changes outside the function usually affect the graph vertically.
  
  Changes inside the function usually affect the graph horizontally.
  
  This is the single most important idea in graph transformations.

  ---

  Vertical Transformations

  📐 Vertical Shift

  For
  
  $\qquad y=f(x)+k$
  
  

  • if $k>0$, the graph shifts upward by $k$

  
  

  • if $k<0$, the graph shifts downward by $|k|$

  
  
  The shape does not change.

  📐 Vertical Scaling

  For
  
  $\qquad y=af(x)$
  
  

  • if $|a|>1$, vertical stretch by factor $|a|$

  
  

  • if $0<|a|<1$, vertical compression by factor $|a|$

  
  

  • if $a<0$, there is also reflection in the $x$-axis

  Example From $y=x^2$:
  • $y=x^2+3$ is shifted up by $3$
  • $y=2x^2$ is vertically stretched
  • $y=-x^2$ is reflected in the $x$-axis
  ---

  Horizontal Transformations

  📐 Horizontal Shift

  For
  
  $\qquad y=f(x-a)$
  
  the graph shifts right by $a$.
  
  For
  
  $\qquad y=f(x+a)$
  
  the graph shifts left by $a$.
  
  This opposite-looking behaviour is one of the most tested traps.

  📐 Horizontal Scaling

  For
  
  $\qquad y=f(bx)$
  
  

  • if $|b|>1$, horizontal compression by factor $\dfrac{1}{|b|}$

  
  

  • if $0<|b|<1$, horizontal stretch by factor $\dfrac{1}{|b|}$

  
  

  • if $b<0$, there is also reflection in the $y$-axis

  ⚠️ Most Important Trap

  Inside changes work in the reverse-looking direction.
  
  So:
  

  • $f(x-3)$ means shift right by $3$

  
  

  • $f(x+3)$ means shift left by $3$

  
  

  • $f(2x)$ means horizontal compression, not stretch

  ---

  Reflections

  📐 Reflection in the $x$-Axis

  For
  
  $\qquad y=-f(x)$
  
  every point $(x,y)$ becomes $(x,-y)$.
  
  So the graph reflects in the $x$-axis.

  📐 Reflection in the $y$-Axis

  For
  
  $\qquad y=f(-x)$
  
  every point $(x,y)$ on $y=f(x)$ becomes $(-x,y)$.
  
  So the graph reflects in the $y$-axis.

  ❗ Coordinate View

  If $(a,b)$ lies on $y=f(x)$, then:
  
  

  • on $y=f(x-h)$, the corresponding point is $(a+h,b)$

  
  

  • on $y=f(x)+k$, the corresponding point is $(a,b+k)$

  
  

  • on $y=-f(x)$, the corresponding point is $(a,-b)$

  
  

  • on $y=f(-x)$, the corresponding point is $(-a,b)$

  ---

  Combined Transformation

  📐 General Form

  A very common form is
  
  $\qquad y=A\,f(B(x-h))+k$
  
  Read it in pieces:
  
  

  • $x-h$ gives horizontal shift

  
  

  • $B$ gives horizontal scaling and possibly reflection in the $y$-axis

  
  

  • $A$ gives vertical scaling and possibly reflection in the $x$-axis

  
  

  • $+k$ gives vertical shift

  💡 Safe Order for Reading

  To interpret
  $\qquad y=A\,f(B(x-h))+k$
  
  it is usually safest to read:
  

  • inside first

  
  

  • outside later

  
  
  That is:
  

  • horizontal effects first

  
  

  • vertical effects after that

  ---

  Modulus Transformations

  📐 $y=|f(x)|$

  Start with the graph of $y=f(x)$.
  
  

  • the part already above the $x$-axis stays unchanged

  
  

  • the part below the $x$-axis is reflected upward into the positive side

  
  
  So the final graph lies on or above the $x$-axis.

  📐 $y=f(|x|)$

  Start with the graph of $y=f(x)$ for $x\ge 0$.
  
  Then reflect that right-half graph in the $y$-axis.
  
  So the final graph is symmetric about the $y$-axis.

  ⚠️ Do Not Mix These Up
  • $|f(x)|$ changes output signs
  • $f(|x|)$ changes the input before the function acts
  These are very different graphs.
  ---

  Standard Parent Graphs

  📐 Graphs You Should Know Well

  • $\qquad y=x$

  
  

  • $\qquad y=x^2$

  
  

  • $\qquad y=x^3$

  
  

  • $\qquad y=|x|$

  
  

  • $\qquad y=\sqrt{x}$
    
    • $\qquad y=\dfrac{1}{x}$
    
    
    • $\qquad y=\sin x$
    
    
    • $\qquad y=\cos x$

  💡 Why Parent Graphs Matter

  Most graph-transformation questions are really asking:
  

  • what happens to a familiar graph

  
  

  • where key points move

  
  

  • whether symmetry, intercepts, or turning points shift

  ---

  Minimal Worked Examples

  Example 1 Describe the graph of $\qquad y=(x-2)^2+3$ relative to $y=x^2$. The $x-2$ shifts the graph right by $2$. The $+3$ shifts it up by $3$. So the parabola moves from vertex $(0,0)$ to vertex $(2,3)$. --- Example 2 Describe the graph of $\qquad y=f(-2x)$ relative to $y=f(x)$. The negative sign inside gives reflection in the $y$-axis. The factor $2$ inside gives horizontal compression by factor $\dfrac{1}{2}$. So the graph is reflected in the $y$-axis and compressed horizontally. ---

  Domain and Range Under Transformations

  ❗ Domain and Range Thinking

  Transformations also affect domain and range.
  
  

  • vertical shift changes the range

  
  

  • horizontal shift changes the domain

  
  

  • vertical scaling changes range values

  
  

  • horizontal scaling changes input positions

  
  

  • reflections may reverse parts of the graph but do not by themselves change domain size or range size

  Example For $\qquad y=\sqrt{x}$ domain is $[0,\infty)$. For $\qquad y=\sqrt{x-3}$ domain becomes $[3,\infty)$. For $\qquad y=\sqrt{x}+3$ domain stays $[0,\infty)$ but range becomes $[3,\infty)$. ---

  Recognition Guide

  💡 Fast Identification
  • $f(x)+k$ $\Rightarrow$ vertical shift
  • $f(x-h)$ $\Rightarrow$ horizontal shift
  • $af(x)$ $\Rightarrow$ vertical scale
  • $f(bx)$ $\Rightarrow$ horizontal scale
  • $-f(x)$ $\Rightarrow$ reflection in $x$-axis
  • $f(-x)$ $\Rightarrow$ reflection in $y$-axis
  • $|f(x)|$ $\Rightarrow$ negative parts go upward
  • $f(|x|)$ $\Rightarrow$ reflect right half into the left half
  ---

  Common Traps

  ⚠️ Avoid These Errors
  • ❌ saying $f(x-2)$ shifts left
  • ❌ saying $f(2x)$ is a horizontal stretch
  • ❌ confusing $|f(x)|$ with $f(|x|)$
  • ❌ applying vertical logic to inside changes
  • ❌ ignoring how domain and range move under transformations
  • ❌ forgetting that negative factors may create reflections
  ---

  CMI Strategy

  💡 How to Solve Graph Transformation Questions

  • start from the parent graph

  
  

  • identify whether each change is inside or outside

  
  

  • track a few key points like intercepts, vertex, endpoint, or turning point

  
  

  • note reflections before sketching

  
  

  • check symmetry and domain/range after transformation

  
  

  • when two modulus operations appear, treat one at a time carefully

  ---

  Practice Questions

  :::question type="MCQ" question="The graph of $y=f(x-4)$ is obtained from the graph of $y=f(x)$ by" options=["shifting left by $4$","shifting right by $4$","stretching horizontally by factor $4$","reflecting in the $y$-axis"] answer="B" hint="Inside shift acts in the reverse-looking direction." solution="In the expression $f(x-4)$, the change is inside the function, so it affects the graph horizontally. The rule is: $\qquad y=f(x-a)$ shifts the graph right by $a$. Hence $y=f(x-4)$ is obtained by shifting the graph right by $4$. Therefore, the correct option is $\boxed{B}$." ::: :::question type="NAT" question="The vertex of the parabola $y=(x-3)^2-5$ is $(a,b)$. Find the value of $a+b$." answer="-2" hint="Compare with the graph of $y=x^2$." solution="The graph of $y=x^2$ has vertex $(0,0)$. In $y=(x-3)^2-5$, the graph is shifted right by $3$ and down by $5$. So the new vertex is $\qquad (a,b)=(3,-5)$ Hence $\qquad a+b=3+(-5)=-2$ Therefore, the answer is $\boxed{-2}$." ::: :::question type="MSQ" question="Which of the following statements are true?" options=["$y=-f(x)$ is the reflection of $y=f(x)$ in the $x$-axis.","$y=f(-x)$ is the reflection of $y=f(x)$ in the $y$-axis.","$y=|f(x)|$ is always above or on the $x$-axis.","$y=f(|x|)$ is always the reflection of the full graph of $f(x)$ in the $x$-axis."] answer="A,B,C" hint="Separate output changes from input changes." solution="1. True. Multiplying the output by $-1$ reflects the graph in the $x$-axis.
  • True. Replacing $x$ by $-x$ reflects the graph in the $y$-axis.
  • True. In $|f(x)|$, all negative output values become positive, so the graph lies on or above the $x$-axis.
  • False. The graph of $f(|x|)$ is obtained by taking the right-half graph and reflecting it in the $y$-axis, not the $x$-axis.
  Hence the correct answer is $\boxed{A,B,C}$." ::: :::question type="SUB" question="Describe how the graph of $y=\sqrt{x}$ changes to become the graph of $y=2\sqrt{x-1}+3$." answer="Shift right by $1$, stretch vertically by factor $2$, then shift up by $3$." hint="Read the inside change first, then the outside changes." solution="Start with the graph $\qquad y=\sqrt{x}$ Now consider $\qquad y=2\sqrt{x-1}+3$ Step 1: The inside change $x-1$ shifts the graph right by $1$. Step 2: The factor $2$ outside gives a vertical stretch by factor $2$. Step 3: The final $+3$ shifts the graph upward by $3$. So the transformation is: $\qquad \boxed{\text{shift right by }1,\ \text{stretch vertically by factor }2,\ \text{then shift up by }3}$" ::: ---

  Summary

  ❗ Key Takeaways for CMI

  • outside changes act vertically, inside changes act horizontally

  
  

  • $f(x-a)$ shifts right, while $f(x+a)$ shifts left

  
  

  • $af(x)$ changes vertical scale, while $f(bx)$ changes horizontal scale

  
  

  • $-f(x)$ reflects in the $x$-axis and $f(-x)$ reflects in the $y$-axis

  
  

  • $|f(x)|$ and $f(|x|)$ are very different transformations

  
  

  • graph questions become easier when you track key points from a known parent graph

  ---

  💡 Next Up

  Proceeding to Modulus functions.

  ---

  Part 4: Modulus functions

  Modulus Functions

  Overview

  Modulus functions are built from the absolute value operation. They are among the most useful objects in graph-based function understanding because they combine algebra, geometry, and piecewise thinking. In CMI-style questions, modulus is rarely just about “making things positive”; it is tested through graphs, transformations, equations, inequalities, symmetry, distance interpretation, and counting intersections. ---

  Learning Objectives

  ❗ By the End of This Topic

  After studying this topic, you will be able to:
  

  • Interpret $|x|$ and $|f(x)|$ both algebraically and geometrically.

  
  

  • Rewrite modulus expressions as piecewise functions.

  
  

  • Sketch graphs of basic and transformed modulus functions.

  
  

  • Solve equations and inequalities involving one or more modulus signs.

  
  

  • Understand symmetry, distance, and graph intersections involving modulus functions.

  ---

  Core Meaning

  📖 Absolute Value

  For any real number $x$,
  
  $\qquad |x| = <br>\begin{cases}<br>x, & x \ge 0 \\ <br>-x, & x < 0 <br>\end{cases}$
  
  So $|x|$ is the non-negative magnitude of $x$.

  ❗ Distance Interpretation

  The quantity $|x-a|$ represents the distance of the point $x$ from the point $a$ on the real line.
  
  Examples:
  

  • $|x|$ = distance from $0$

  
  

  • $|x-3|$ = distance from $3$

  
  

  • $|x+2|$ = distance from $-2$

  This interpretation is extremely useful for solving equations and inequalities quickly. ---

  Basic Properties

  📐 Standard Properties of Modulus

  For all real numbers $x,y$:
  
  

  • $|x| \ge 0$

  
  

  • $|x| = 0 \iff x=0$

  
  

  • $|-x| = |x|$

  
  

  • $|xy| = |x||y|$

  
  

  • $\left|\dfrac{x}{y}\right| = \dfrac{|x|}{|y|}$ for $y \ne 0$
    
    • $|x|^2 = x^2$
    
    
    • $\sqrt{x^2} = |x|$

  📐 Two Very Useful Identities
  • $\max(x,-x) = |x|$
  • $|a-b|$ is the distance between $a$ and $b$
  ---

  Piecewise Form

  📐 How to Open a Modulus

  To remove modulus signs, find where the inside changes sign.
  
  For example,
  
  $\qquad |x-2| = <br>\begin{cases}<br>x-2, & x \ge 2 \\ <br>2-x, & x < 2 <br>\end{cases}$
  
  Similarly,
  
  $\qquad |2x+1| = <br>\begin{cases}<br>2x+1, & x \ge -\dfrac{1}{2} \\ <br>-(2x+1), & x < -\dfrac{1}{2} <br>\end{cases}$

  💡 Golden Rule

  Do not open a modulus sign blindly. First find where the inner expression is zero. That point splits the real line into intervals.

  ---

  Graph of $y=|x|$

  📖 Basic Modulus Graph

  The graph of
  
  $\qquad y=|x|$
  
  is V-shaped, with:
  

  • vertex at $(0,0)$

  
  

  • slope $-1$ on the left

  
  

  • slope $+1$ on the right

  
  

  • symmetry about the $y$-axis

  ❗ Piecewise Graph Form

  $\qquad y=|x| = <br>\begin{cases}<br>x, & x \ge 0 \\ <br>-x, & x < 0 <br>\end{cases}$
  
  So the graph is made of two straight lines meeting at the origin.

  ---

  Transformations of Modulus Graphs

  📐 General Form

  A very common family is
  
  $\qquad y=a|x-h|+k$
  
  This is the graph of $|x|$ transformed as follows:
  

  • shift right by $h$

  
  

  • shift up by $k$

  
  

  • vertical stretch by factor $|a|$

  
  

  • reflection in the $x$-axis if $a<0$

  
  
  The vertex is at
  
  $\qquad (h,k)$

  Examples:
  • $y=|x-3|$ has vertex $(3,0)$
  • $y=|x|+2$ has vertex $(0,2)$
  • $y=-|x+1|+4$ has vertex $(-1,4)$ and opens downward
  • $y=2|x-5|-1$ has vertex $(5,-1)$ and steeper arms
  :::
  💡 Graph-Based Interpretation

  The graph of $y=|f(x)|$ is obtained by:
  

  • keeping the part of $y=f(x)$ above the $x$-axis unchanged

  
  

  • reflecting the part below the $x$-axis upward across the $x$-axis

  💡 Graph of $y=f(|x|)$

  The graph of $y=f(|x|)$ is obtained by:
  

  • taking the graph of $f(x)$ for $x \ge 0$

  
  

  • reflecting it across the $y$-axis

  These two transformations are tested very often. ---

  Algebraic Forms You Must Know

  📐 Useful Rewrites
  • $|x| =
  \begin{cases} x,& x\ge 0\\ -x,& x<0 \end{cases}$
  • $|x-a| =
  \begin{cases} x-a,& x\ge a\\ a-x,& x
  • $|ax+b|$ changes form at
  $\qquad ax+b=0$
  • $|x|^2=x^2$
  • $\sqrt{x^2}=|x|$