Dr. Math Azzam Platform | High School & University Mathematics

Explanation

01 · See the starting shape

Parent Functions & Guide Points

Recognize the function family before any transformation: linear, quadratic, cubic, absolute value, square root, or reciprocal. A guide point is a reliable point on the original graph; transformed points must be computed, not guessed.

Linear

f(x)=x

Guide points: (-1, -1) · (0, 0) · (1, 1)

Quadratic

f(x)=x^2

Guide points: (-2, 4) · (0, 0) · (2, 4)

Cubic

f(x)=x^3

Guide points: (-1, -1) · (0, 0) · (1, 1)

Absolute value

f(x)=|x|

Guide points: (-1, 1) · (0, 0) · (1, 1)

Square root

f(x)=\sqrt{x}

Guide points: (0, 0) · (1, 1) · (4, 2)

Reciprocal

f(x)=\frac{1}{x}

Guide points: (-1, -1) · (1, 1)

Explanation

Mr. Azzam's clue · Always start with the parent

Write the rule, choose recognizable points, check asymptotes or restricted domains, then transform all required points with exactly the same operation.

Explanation

02 · Read the inside and outside

HSRV — Your Transformation Roadmap

Use the HSRV framework: Horizontal shifting, Stretching/shrinking, Reflecting, Vertical shifting. Each can be horizontal or vertical where applicable.

Explanation

Mr. Azzam's clue · Outside: keep it · Inside: inverse it

Outside the function, operations change y-values directly. Inside the function, operations change x-values inversely. A minus sign outside reflects across the x-axis; a minus sign inside reflects across the y-axis.

g(x)=a f(bx-h)+k,\qquad b\ne0

(x,y)\mapsto\left(\frac{x+h}{b},\;ay+k\right)

Solve the inside equation for the new input: $bX-h=x$ . In this unfactored form, the horizontal shift is $H=h/b$ , not generally $h$ .

bx-h=b(x-H),\qquad H=\frac{h}{b}

The equivalent factored form is $a f(b(x-H))+k$ , with point map $(x,y)\mapsto(H+x/b,ay+k)$ . The lab uses its $h$ control in $bx-h$ , so the actual horizontal shift is $H=h/b$ when $b\ne0$ .

For a transformed domain, solve $bx-h\in D_f$ . When dividing an inequality by a negative $b$ , reverse its direction.

New function	Effect	Guide point
$f(x)+3$	up 3	(x,y+3)
$f(x-2)$	right 2	(x+2,y)
$-f(x)$	reflect in x-axis	(x,-y)
$f(-x)$	reflect in y-axis	(-x,y)
$2f(x)$	vertical stretch 2	(x,2y)
$f(2x)$	horizontal shrink to half	(x/2,y)

Your turn

Question 1 of 2

What happens to the point

(3,2)

under

g(x)=f(x+4)-1

?

(-1,1)

(7,1)

(-1,3)

(7,3)

Interactive lab

03 · Interactive exploration

Predict, Move, and Verify with HSRV

Choose a family, then modify horizontal shifts, vertical shifts, reflections, and stretch factors. Use guide points before clicking Check.

Interactive lab

HSRV · Interactive Transformation Lab

Predict → Transform → Examine guide points → Explain

Parent functions · Absolute Value

● Parent (dashed) · ● Transformed

Guide points

Parameters & HSRV

a

b

h

k

HSRV, domain & range

HNo horizontal shift

SVertical ×1; horizontal ×1

RNo reflections.

VNo vertical shift

Domain

\mathbb{R}

Range

[0,\infty)

Transformation stages

Horizontal scale/reflection → vertical scale/reflection → horizontal translation → vertical translation

Mr. Azzam’s clue

Mr. Azzam’s clue: outside changes y directly; inside changes x inversely.

\boxed{(x,y)\longmapsto\left(\frac{x+h}{b},ay+k\right)}\qquad b\ne0

Click a transformed point to inspect its exact mapping. When b = 0, the graph uses the constant input −h only if f(−h) exists.

Worked example

04 · Worked progression

My Guide-Point Routine

Worked example

Worked example · From a parent point to a new point

Suppose the parent graph contains $(4,2)$ and $g(x)=-2f(x-3)+1$ .

1. Input changes horizontally: 4 becomes 7 (three right).

2. Output changes vertically: 2 becomes −4, then rises to −3.

Final point: $(7,-3)$ .

Check directly from the map: $(x,y)\mapsto(x+3,-2y+1)$ .

Worked example

Worked example · Square-root translation, using all guide points

For $f(x)=\sqrt{x}$ , use (0,0), (1,1), (4,2), (9,3).

$g(x)=\sqrt{x-2}+3$ moves every input 2 right and every output 3 up.

The mapped points are (2,3), (3,4), (6,5), (11,6). The graph begins at (2,3) and its domain is $[2,\infty)$ .

f(x) = √x
g(x) = √(x − 2) + 3

Your turn

Question 1 of 2

The parent point

(1,2)

is transformed by

g(x)=3f(2(x-4))-1

. Find the image point.

(6,5)

(4.5,7)

(3,5)

(4.5,5)

Practice

05 · Hands-on transformation

Graph-to-Graph Practice Lab

Use the five given guide points to sketch each of six transformations before revealing the completed graph.

Interactive lab

Plot → Connect → Check

Plot all five transformed guide points on the grid; connect in original point order.

0/5 points

Original points

(0, 1) · (1, 2) · (3, 0) · (4, -1) · (6, 0)

● Blue parent ● Green your points ● Orange solution

How to plot and connect

Select five transformed points in left-to-right original order. Click to add a point, then drag green points to correct them. Connect the five points in parent-point order.

Interactive lab

Draw It Yourself · Graph Practice

Eight original graphical tasks. The blue graph is the given function. Work out the transformed guide point and sketch the orange curve on paper before checking.

Graph problem 1 / 8 · Correct 0 / 0 attempted

Sketch

y=g(x)+2

from the given graph of g.

Highlighted parent point (1, 2) — where does it move?

Mr. Azzam's First Move

Separate inside changes to x from outside changes to y. Calculate the image of the highlighted point before sketching.

Worked example

06 · Reverse translation & composition

What Does the Equation Tell You?

Worked example

Worked example · Describe the combined transformation

$y=|x+3|-6$ comes from $y=|x|$ : 3 units left and 6 units down. The vertex moves from (0,0) to (−3,−6).

y = |x|
y = |x + 3| − 6

Worked example

Worked example · Write the equation after a shift

Starting with $y=x^2$ , shift 7 units right and 2 units down: $y=(x-7)^2-2$ .

y = x²
y = (x − 7)² − 2

Worked example

Worked example · Distinguish stretching from shifting

$y=\sqrt{x+2}+2$ shifts left 2 and up 2. But $y=2\sqrt{x}$ stretches vertically by a factor of 2; its endpoint remains (0,0).

y = √(x + 2) + 2
y = 2√x

Your turn

Question 1 of 2

Which graph is 2 units to the left of

y=\sqrt{x}

?

Parent y = √x

Use the parent graph to identify the requested translation.

y=\sqrt{x}+2

y=2\sqrt{x}

y=\sqrt{x+2}

y=\sqrt{x-2}

Explanation

07 · Symmetry connection

Even, Odd or Neither?

A graph is even if it is symmetric about the y-axis, and odd if it is symmetric about the origin. The domain must also support the required symmetry.

f(-x)=f(x)\quad\text{even}

f(-x)=-f(x)\quad\text{odd}

Explanation

Mr. Azzam's clue · Substitute −x first

Replace every x by (−x), simplify powers and signs, then compare with the original f(x) and its negative −f(x). Never decide solely from a single plotted point.

Interactive lab

Even, Odd or Neither? · Symmetry Lab

Compare the outputs at x and -x. Predict the symmetry, then reveal the algebraic test.

Choose x = 1.0

$f(x)$ = 2.000; $f(-x)$ = -2.000

Your turn

Question 1 of 2

Classify

f(x)=x^5+x

.

BothOddEvenNeither

Key takeaway

08 · Check every transformation

Exam Strategy and Mastery

Key takeaway

The Mr. Azzam checklist

Recognize the parent and mark guide points.
Read inside versus outside correctly.
Follow HSRV and write the image of every point.
Plot the final graph with an accurate coordinate scale.
Check domain, range, endpoints and asymptotes after transformation.
For symmetry, verify f(−x) algebraically.

Lesson Mastery Checkpoint

Rate each skill. This is only the starting signal; your actual Practice and Assessment performance will update the plan.

Parent function guide points

Horizontal and vertical shifts

Horizontal and vertical stretch/shrink

Reflections in axes

HSRV and point mapping

Graph-to-graph transformations

Write equations from transformations

Even and odd functions

Rate all skills to prepare your personalized practice path.

Find your next lesson

Transformations of Functions

Parent Functions & Guide Points

HSRV — Your Transformation Roadmap

Your turn

Your turn · student questions

Predict, Move, and Verify with HSRV

HSRV · Interactive Transformation Lab

Parameters & HSRV

My Guide-Point Routine

Your turn

Your turn · student questions

Graph-to-Graph Practice Lab

Plot → Connect → Check

Draw It Yourself · Graph Practice

What Does the Equation Tell You?

Your turn

Your turn · student questions

Even, Odd or Neither?

Even, Odd or Neither? · Symmetry Lab

Your turn

Your turn · student questions

Exam Strategy and Mastery

Keep your unfinished answers?