Graphs and Transformations.

Grade 12 Mathematics: Section 1.4 - Graphs and Transformations: Functions, Transformations, Trigonometric Graphs

Subtopic Navigator

Introduction to Graph Transformations
Parent Functions & Key Features
Vertical & Horizontal Translations
Reflections (Vertical & Horizontal)
Stretches & Compressions
Methods & Techniques (Order of Transformations)
Common Mistakes & Misconceptions
Real-World Applications (Modeling)
Practice Exercises (Including Trigonometric Graphs)
Conclusion & Summary

Lesson Objectives

Identify parent functions: linear, quadratic, cubic, reciprocal, trigonometric (sin, cos, tan).
Apply translations, reflections, and stretches to graphs of functions.
Write transformed equations in the form $y = a f(b(x-h)) + k$.
Analyze the effect of parameters on trigonometric graphs: amplitude, period, phase shift, vertical shift.
Sketch transformed graphs accurately and determine key points (intercepts, turning points, asymptotes).
Combine multiple transformations in the correct order.

Introduction to Graph Transformations

In Grade 12, understanding how to manipulate graphs is essential for calculus, physics, and applied mathematics. Transformations allow us to move, stretch, or reflect any parent function. The general transformed function is $y = a f(b(x-h)) + k$, where each parameter has a geometric meaning. We also apply these ideas to trigonometric graphs, which are fundamental for modeling periodic phenomena.

General Transformation Form
$y = a f(b(x-h)) + k$
• $a$ : vertical stretch ($|a|>1$) or compression ($0<|a|<1$); if $a<0$: reflection over x-axis.
• $b$ : horizontal stretch/compression (period change for trig); if $b<0$: reflection over y-axis.
• $h$ : horizontal shift (opposite sign: $x-h$ means shift right by $h$).
• $k$ : vertical shift (up if $k>0$, down if $k<0$).

Key Definitions:
• Parent Function: The simplest form of a function (e.g., $y=x^2$, $y=\sin x$, $y=|x|$).
• Translation: Shifting a graph horizontally or vertically without changing shape.
• Reflection: Flipping a graph over an axis (x-axis or y-axis).
• Stretch/Compression: Multiplying coordinates vertically or horizontally.
• Amplitude (Trig): Half the vertical distance between max and min of sine/cosine, $|a|$.
• Period (Trig): Horizontal length of one cycle: $\frac{2\pi}{|b|}$ for sin/cos; $\frac{\pi}{|b|}$ for tan.

Parent Functions & Key Features

Before transforming, we must know the basic shape of common parent functions. Recognizing these helps predict how transformations affect the graph.

Essential Parent Functions for Grade 12:
1. Linear: $f(x)=x$ (diagonal line through origin)
2. Quadratic: $f(x)=x^2$ (parabola, vertex at origin)
3. Cubic: $f(x)=x^3$ (point of inflection at origin)
4. Reciprocal: $f(x)=\frac{1}{x}$ (two branches, asymptotes at x=0, y=0)
5. Sine: $f(x)=\sin x$ (period $2\pi$, amplitude 1, range [-1,1])
6. Cosine: $f(x)=\cos x$ (similar to sine, phase shift)
7. Tangent: $f(x)=\tan x$ (period $\pi$, asymptotes at $x=\frac{\pi}{2}+n\pi$)

Parent Function Summary Table

Function Name	Parent Equation	Key Features
Linear	$f(x)=x$	Slope 1, passes through (0,0)
Quadratic	$f(x)=x^2$	Vertex (0,0), axis of symmetry x=0
Cubic	$f(x)=x^3$	Increasing, rotational symmetry about origin
Reciprocal	$f(x)=\frac{1}{x}$	Asymptotes x=0, y=0
Sine	$f(x)=\sin x$	Period $2\pi$, amplitude 1, zeros at $n\pi$
Cosine	$f(x)=\cos x$	Period $2\pi$, amplitude 1, max at $x=0$
Tangent	$f(x)=\tan x$	Period $\pi$, asymptotes at $\pi/2 + n\pi$

Watch Out!
For horizontal shifts, the sign inside the function is opposite: $f(x-h)$ shifts RIGHT by $h$, while $f(x+h)$ shifts LEFT.

Practice for Concept 1 (Identifying Parent Functions)

What is the parent function of $y = 3(x-2)^2 + 1$?
What is the parent function of $y = -2\sin(3x) + 4$?
Describe the shape of $y = \frac{1}{x-1} + 2$.

Vertical & Horizontal Translations

Translations move the graph without changing its shape. Vertical shifts add $k$ to the function; horizontal shifts replace $x$ with $(x-h)$.

Example 1: Vertical Translation
Problem: Sketch $y = x^2 + 3$ relative to $y=x^2$.

Solution: The parabola shifts up by 3 units. Vertex moves from (0,0) to (0,3). All y-values increase by 3.

Example 2: Horizontal Translation
Problem: Sketch $y = (x-4)^2$.

Solution: Shift the parabola right by 4 units. Vertex at (4,0).

Example 3: Combined Translation (Trig)
Problem: Write equation for sine curve shifted left $\frac{\pi}{2}$ and down 1.

Solution: $y = \sin(x + \frac{\pi}{2}) - 1$.

Practice Translations

If $f(x)=x^3$, write the equation shifted up 5 and left 2.
Describe the transformation of $y = \cos x$ to $y = \cos(x - \pi) + 2$.

Reflections (Vertical & Horizontal)

A negative sign outside the function reflects over the x-axis; a negative sign inside (on the input) reflects over the y-axis.

Example 1: Reflection over x-axis
Problem: Graph $y = -\sqrt{x}$ compared to $y=\sqrt{x}$.

Solution: Each y-value becomes its opposite. The graph flips vertically across the x-axis.

Example 2: Reflection over y-axis
Problem: Graph $y = 2^{-x}$ as a transformation of $y=2^x$.

Solution: Replace $x$ with $-x$, reflecting the exponential graph over the y-axis.

Common Pitfall:
Reflection over y-axis: $f(-x)$ is NOT the same as $-f(x)$. For trigonometric functions, $\sin(-x) = -\sin x$ (odd function), but $\cos(-x)=\cos x$ (even function).

Stretches & Compressions

Vertical stretch: multiply function by $a$ ($|a|>1$ stretch, $0<|a|<1$ compression). Horizontal stretch: replace $x$ with $x/b$ (or write $b$ inside: $f(bx)$ compresses horizontally if $|b|>1$).

Trigonometric Graph Parameters:
For $y = a\sin(b(x-h)) + k$ or $y = a\cos(b(x-h)) + k$:
• Amplitude = $|a|$
• Period = $\frac{360^\circ}{|b|}$ (degrees) or $\frac{2\pi}{|b|}$ (radians)
• Phase shift = $h$ (right if $h>0$)
• Vertical shift = $k$
For $y = a\tan(b(x-h)) + k$: Period = $\frac{\pi}{|b|}$, no amplitude (unbounded).

Example 1: Vertical Stretch & Horizontal Compression
Problem: Describe transformations: $y = 3\sin(2x)$.

Solution: Amplitude = 3 (vertical stretch by 3), period = $\frac{2\pi}{2} = \pi$ (horizontal compression by factor 1/2).

Example 2: Horizontal Stretch
Problem: Write equation for cosine with period $4\pi$ and amplitude 0.5.

Solution: $y = 0.5\cos\left(\frac{1}{2}x\right)$ because $b = \frac{1}{2}$ gives period $2\pi/(1/2)=4\pi$.

Effect of Parameter $b$ on Period of $\sin(bx)$

$b$ value	Period	Effect
$1$	$2\pi$	Standard period
$2$	$\pi$	Compressed horizontally (more waves)
$\frac{1}{2}$	$4\pi$	Stretched horizontally (fewer waves)

Practice for Trigonometric Transformations

Find amplitude, period, phase shift of $y = 4\cos(3x - \pi) + 1$.
Write equation of tangent function with period $\frac{\pi}{2}$ and vertical shift -2.
Graph one period of $y = -2\sin(\frac{1}{2}x)$.

Methods & Techniques (Order of Transformations)

When multiple transformations apply, the correct order is: horizontal stretches/reflections (inside), then horizontal shifts, then vertical stretches/reflections, then vertical shifts.

Step-by-Step Transformation Order:
Starting from $y = f(x)$ to $y = a f(b(x-h)) + k$:
1. Horizontal stretch/compression by factor $1/b$ (if $b$ negative, reflect over y-axis).
2. Horizontal shift by $h$ (right if $h>0$).
3. Vertical stretch/compression by factor $a$ (if $a$ negative, reflect over x-axis).
4. Vertical shift by $k$ (up if $k>0$).
Tip: When sketching, identify key points of parent function and apply transformations in this order.

Example: Order of Transformations
Problem: Describe steps to graph $y = -2\sqrt{x-3} + 1$ from $y=\sqrt{x}$.

Solution:
1. Horizontal shift right 3: $\sqrt{x} \to \sqrt{x-3}$.
2. Vertical stretch by factor 2: $\sqrt{x-3} \to 2\sqrt{x-3}$.
3. Reflection over x-axis (negative): $2\sqrt{x-3} \to -2\sqrt{x-3}$.
4. Vertical shift up 1: $-2\sqrt{x-3} \to -2\sqrt{x-3}+1$.

Trigonometric Example: Combining Transformations
Problem: Write $y = 3\tan(2x + \pi) - 4$ in the form $y = a\tan(b(x-h)) + k$. Then describe transformations.

Solution: Factor: $y = 3\tan(2(x + \frac{\pi}{2})) - 4$. So $a=3, b=2, h=-\frac{\pi}{2}, k=-4$.
Transformations: horizontal compression by 1/2, horizontal shift left $\pi/2$, vertical stretch by 3, vertical shift down 4.

Common Mistakes & Misconceptions

Pitfall 1: Horizontal shift direction. $y = f(x-3)$ shifts RIGHT, not left. Remember: "minus means right".
Pitfall 2: Order of reflections and stretches. Always do stretches before translations, but horizontal before vertical.
Pitfall 3: Confusing amplitude with vertical stretch. For $\sin$ and $\cos$, amplitude is $|a|$, but for $\tan$ there is no amplitude.
Pitfall 4: Period calculation. Period of $\sin(bx)$ is $\frac{2\pi}{|b|}$; forgetting absolute value leads to sign errors.
Pitfall 5: Factoring for phase shift. For $y = \sin(2x - \pi)$, factor: $\sin(2(x - \frac{\pi}{2}))$, phase shift is $\frac{\pi}{2}$ right.

Real-World Applications (Modeling)

Transformations are used to model real phenomena: tides, sound waves, seasonal temperatures, and engineering design.

Application 1: Tides
Problem: In a harbor, water depth varies as a cosine function: depth $d(t) = 4.5\cos(0.5(t-4)) + 8$ meters, $t$ in hours. Find amplitude, period, and vertical shift.

Solution: Amplitude = 4.5 m, period = $\frac{2\pi}{0.5} = 4\pi \approx 12.57$ hours, vertical shift = 8 m (mean depth).

Application 2: Pendulum Motion
Scenario: A pendulum's horizontal displacement is $x(t) = 0.2\sin(3t)$ meters. Transformations: amplitude 0.2, period $\frac{2\pi}{3}$ seconds.

Application 3: Quadratic Path (Projectile)
Problem: A ball thrown follows $h(t) = -5(t-2)^2 + 20$ meters. Vertex at (2,20) shows max height 20m at 2 seconds.

Cumulative Practice Exercises (Functions & Trig Graphs)

Attempt all. Show steps. Use the transformation order.

Given $f(x)=x^2$, write the equation after a vertical stretch by 3, reflection over x-axis, shift left 4 and up 2.
Find amplitude, period, phase shift, vertical shift of $y = -5\sin(4x + \pi) - 2$.
Sketch the graph of $y = \frac{1}{x-2} - 1$, showing asymptotes.
If the point $(3, 5)$ lies on $y = f(x)$, where does it move under $y = 2f(x-1) + 3$?
Write a cosine function with amplitude 3, period $90^\circ$, phase shift $30^\circ$ right, vertical shift down 1.
Describe transformations to obtain $y = -\tan\left(\frac{x}{2} + \frac{\pi}{4}\right)$ from $y=\tan x$.
Graph one full period of $y = 2\cos\left(x - \frac{\pi}{2}\right)$.
Error analysis: A student claims $y = \sin(2x - \pi)$ has phase shift $\pi$. Correct the student.
Determine the equation of the quadratic function with vertex $(-2,5)$ passing through $(0,1)$.
For $g(x) = 3 - 4\sin(0.5x)$, find max and min values, and the first positive x-intercept.

Show/Hide Answers

Answers to Cumulative Exercises

Problem 1: $y = -3(x+4)^2 + 2$.
Problem 2: Amplitude = 5, Period = $\frac{2\pi}{4} = \frac{\pi}{2}$, Phase shift: factor $4(x + \pi/4)$ → shift left $\pi/4$, vertical shift = -2.
Problem 3: Vertical asymptote $x=2$, horizontal asymptote $y=-1$.
Problem 4: $x' = 3+1 = 4$, $y' = 2(5)+3 = 13$ → $(4,13)$.
Problem 5: $y = 3\cos(4(x - 30^\circ)) - 1$ (since period $90^\circ = \frac{\pi}{2}$ rad → $b = 360/90 = 4$).
Problem 6: $y = -\tan\left(\frac{1}{2}(x + \frac{\pi}{2})\right)$. Transformations: horizontal stretch by 2, left shift $\pi/2$, reflection over x-axis.
Problem 7: Cosine shifted right $\pi/2$, amplitude 2, period $2\pi$ → graph starts at max at $\pi/2$.
Problem 8: Correction: $y = \sin(2(x - \pi/2))$, phase shift = $\pi/2$ right, not $\pi$.
Problem 9: $y = a(x+2)^2 + 5$; using (0,1): $1 = 4a + 5$ → $a = -1$, so $y = -(x+2)^2 + 5$.
Problem 10: Range: $[-1, 7]$ (max 7, min -1). Set $3 - 4\sin(0.5x)=0$ → $\sin(0.5x)=0.75$ → $0.5x = \arcsin(0.75) \approx 0.848$ → $x \approx 1.696$.

Conclusion & Summary

Graph transformations allow us to move and reshape any function, which is critical for modeling real-world data. Mastering $y = a f(b(x-h)) + k$ gives you control over amplitude, period, phase shift, and vertical translation — especially for trigonometric functions. Always apply horizontal transformations before vertical ones, and remember the reverse sign for horizontal shifts.

Key Takeaways:
1. Parent functions provide the basic graph shape.
2. $a$: vertical stretch/reflection, $b$: horizontal stretch/compression/reflection, $h$: horizontal shift, $k$: vertical shift.
3. For sine/cosine: amplitude $|a|$, period $2\pi/|b|$, phase shift $h$.
4. For tangent: period $\pi/|b|$, vertical asymptotes shift with phase shift.
5. Order of operations: inside (horizontal) then outside (vertical).
6. Use key points (intercepts, maxima, minima, asymptotes) to sketch accurately.

Continue practicing with past exam questions — transformations are a favorite in Grade 12 final exams!