Dr. Math Azzam Platform | High School & University Mathematics

Explanation

01

Recognize before you sketch

Graphing Toolkit · Start from the Family

Start by reconnecting equations with familiar graph shapes. Before plotting many points, identify the family and the information that controls its shape.

Explanation

y=2x-1

Linear · slope 2 · y-intercept −1

Explanation

y=x^2

Quadratic · vertex (0,0)

Explanation

y=(x-2)^2+3

Quadratic · vertex (2,3)

Explanation

y=x^2-4x+3

Quadratic · standard form; find key points

Explanation

y=|x-2|+3

Absolute value · V-shape · vertex (2,3)

Worked example · translated quadratic

Sketch

y=(x+1)^2-4

.

Analyze first; the example is not complete until the graph is drawn.

Your turn · Recognize the family

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Interactive lab

Verify · Translated Quadratic

After sketching by hand, compare your vertex, axis, intercepts, and shape with the exact graph.

Exact mathematical models; sampled curves use independent x/y scales. Toggle an equation to show or hide it.

Optional Desmos exploration

The native graph and checks work without this external service.

Where is the translated parabola’s vertex?

Your turn

What is the first feature to read from

y=|x-4|+1

?

Vertex

(4,1)

Slope 4x-intercept 1A circle center

Explanation

02

Compare shapes

Power Functions · Even, Odd, and the Effect of n

Interactive lab

Power Function Explorer

Compare even and odd powers. Use the graph and the value at x = −1 to identify its symmetry.

Exponent n = 2

Rule: y = x². Landmarks: (−1, 1), (0, 0), (1, 1).

Which symmetry does this power have?

As n increases, the graph becomes flatter near zero and steeper when |x| > 1. Both axes use the same unit length.

Interactive lab

Desmos · Power Family Comparison

Turn expressions on and off to compare even and odd powers on the same axes.

Exact mathematical models; sampled curves use independent x/y scales. Toggle an equation to show or hide it.

Optional Desmos exploration

The native graph and checks work without this external service.

Which symmetry statement is correct?

Key takeaway

Key pattern: even powers have the same general U-shaped behavior as a parabola; odd powers have the same general S-shaped behavior as a cubic. As the exponent increases, the graph becomes flatter near zero and steeper once

|x|>1

.

Worked example · classify before graphing

Compare the general shapes of

f(x)=x^6

and

g(x)=x^5

.

Use parity, not a table of dozens of points.

Your turn · x⁶

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

Question 1 of 2

Which statement is true for

y=x^7

?

It is y-axis symmetricBoth ends riseIt is a step functionIt has odd-power behavior

Explanation

03

One function · different rules

Piecewise Functions · Choose the Interval First

Interactive lab

Piecewise endpoint decisions

Decide the interval first, then use its rule. At a boundary, ≤ or ≥ owns the closed point; < or > gives an open point.

f(x)=\begin{cases}-x,&x\le0\\9-x^2,&0<x\le3\\x-3,&x>3\end{cases}

Input x = 0

Compare the boundary with its neighbors: −0.5, 0, 0.5 and 2.5, 3, 3.5.

Which rule owns x = 0?

Predict f(0).

Worked example · three pieces

Graph

f(x)=\begin{cases}-x,&x\le0\\9-x^2,&0<x\le3\\x-3,&x>3\end{cases}

.

For each rule: identify its interval, graph only that part, then mark its endpoint correctly.

Your turn · Piece 1

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Final graph

Completed graph · three pieces

Worked example · absolute value and root pieces

Graph

f(x)=\begin{cases}1-|x-1|,&x\le2\\\sqrt{x-2},&x>2\end{cases}

.

Do not graph the whole parent function; restrict each rule to its stated interval.

Your turn · First rule

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Final graph

Completed graph · absolute value and root pieces

Common mistake

Common mistake

Drawing each formula for all real x instead of clipping it to its interval.

Your turn

At

x=3

, which condition owns the point:

0<x\le3

or

x>3

?

BothNeither

0<x\le3

x>3

Explanation

04

Find the breaking point

Absolute Value → Piecewise

The central routine is to find where the expression inside the absolute value changes sign. Solve the inside equal to zero, split the number line there, then remove the absolute value with the correct sign.

|u|=\begin{cases}u,&u\ge0\\-u,&u<0\end{cases}

Understanding example · |x|

Rewrite

f(x)=|x|

as a piecewise function.

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · For x≥0

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Final graph

Completed graph · f(x)=|x|

Worked example · shifted absolute value

Rewrite

f(x)=|x-3|

as piecewise.

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Breaking point

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Final graph

Completed graph · f(x)=|x−3|

Worked example · reversed inside

Rewrite

f(x)=|1-x|

as piecewise.

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Breaking point

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Final graph

Completed graph · f(x)=|1−x|

Practice

|2x-1|

Find all breaking point(s), make sign intervals, then write the piecewise rule.

Practice

|2-x|

Find all breaking point(s), make sign intervals, then write the piecewise rule.

Practice

Challenge

|2-x|+1

Find all breaking point(s), make sign intervals, then write the piecewise rule.

Practice

Challenge

|x^2-1|

Find all breaking point(s), make sign intervals, then write the piecewise rule.

Advanced worked example

Rewrite

f(x)=|x^2+x-6|+2x

as a piecewise function.

A quadratic inside absolute value can have more than one breaking point.

Your turn · Factor inside

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

What is the first algebraic step for

|3x+6|

?

DifferentiateSolve

3x+6=0

Square the expressionSet x=0 only

Explanation

05

Reverse the process

Graph → Equation

When a graph is given, identify the family first. For an absolute-value graph, read the vertex and use another point to determine the vertical scale.

y=a|x-h|+k

Vertex $(h,k)$ → substitute one more point → solve for $a$ .

Given graph

Worked example · equation from a V-graph

An absolute-value graph has vertex

(4,1)

and passes through

(6,5)

. Find its equation.

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Start with vertex form

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

A V-graph has vertex

(-2,3)

. Which skeleton is correct?

Illustrative V-graph with the given vertex; steepness is not specified.

y=a|x+2|+3

y=a|x-2|+3

y=a|x+3|-2

y=ax^2+2

Explanation

06

Step by step

Greatest Integer Functions

Interactive lab

Greatest Integer Step Explorer

Predict the greatest integer less than or equal to x. For negative inputs, move down the number line—not toward zero.

Input x = -0.7

The output k satisfies k ≤ x < k + 1. At an integer, the output is that integer itself.

Predict ⌊-0.7⌋.

Key takeaway

Mr. Azzam’s graphing routine

Find the subinterval length from the coefficient of x.
Start subintervals from the zeros/boundaries of the inside expression.
Place equality on the correct endpoint according to the direction of the x coefficient.
Substitute the included endpoint to find the constant value on each step.

Worked example · translated floor function

Sketch

f(x)=\lfloor x+2\rfloor

.

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Step length

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Final graph

Completed graph · floor(x+2)

Worked example · wider steps

Sketch

f(x)=\lfloor \frac{x}{2}+1\rfloor+1

.

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Coefficient of x

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Final graph

Completed graph · wider steps

Worked example · negative coefficient and restricted domain

Sketch

f(x)=\lfloor-2x-1\rfloor

,

x\in[-1,1)

.

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Step length

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Final graph

Completed graph · negative coefficient on restricted domain

Key takeaway

Domain is all real numbers unless restricted. For the basic greatest-integer function, the range is all integers and the graph is constant on each step interval.

Your turn

Question 1 of 2

What is

\lfloor-0.7\rfloor

?

0

1

-0.7

-1

Explanation

07

A visual decision

Vertical Line Test

Interactive lab

Vertical Line Test Lab

Predict how many points share this x-value, then check. A single line with two intersections disproves being a function; one passing line cannot prove it.

Solid: relation. Dashed: test line. Window only; parabolas continue beyond it. Circle radius 2.5. Equal axis units.

y = x²/4 − 2

Vertical line x = 0.8

How many intersections at this x?

A graph represents y as a function of x if and only if no vertical line intersects the graph more than once.

Your turn

A vertical line crosses a graph at two points. What can you conclude?

Illustrative graph matching the stated information

It must be linearIt has no domainThe graph is not y as a function of xIt is always a function

Explanation

08

Algebra can decide too

Does an Equation Define y as a Function of x?

Solve for $y$ . The question is whether each allowed x produces exactly one y-value.

Worked example · function

Does

y-x^2=2

define y as a function of x?

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Solve for y

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Worked example · not a function

Does

y^2+x^2=4

define y as a function of x?

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Isolate y²

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Worked example · rational function

Does

x^2y+y=1

define y as a function of x?

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Factor y

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Worked example · absolute value of y

Does

2x-|y|=0

define y as a function of x?

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Isolate

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Common mistake

Common mistake

Seeing a ± sign and ignoring that it represents two possible outputs. The function question is about the number of y-values for one x.

Your turn

Does

3x-y=5

define y as a function of x?

Only when x>0Yes,

y=3x-5

No, because x and y are both presentNo, because it is linear

Key takeaway

09

Connect everything

Exam Strategy · Analyze → Plot → Sketch → Verify

Key takeaway1 · Analyze

Identify the family, restrictions, breakpoints, vertex, or step width.

Key takeaway2 · Plot & Sketch

Plot the controlling points/endpoints, then complete the actual graph. A graphing question is not finished before the graph.

Key takeaway3 · Verify

Use key points, endpoints, restrictions and Desmos verification to check—not replace—your hand-built graph.

Lesson Mastery Checkpoint

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

Graph families & power functions

Piecewise graphs & endpoints

Absolute value ↔ piecewise

Graph → equation

Greatest integer functions

Vertical line test

Equation → function or not

Rate all skills to prepare your personalized practice path.

Graphing Toolkit · Start from the Family

1. Recognize the family

2. Read the vertex and axis

3. Plot symmetric points

4. Final graph

Your turn

Your turn · student questions

Power Functions · Even, Odd, and the Effect of n

1. x⁶

2. x⁵

3. Compare near zero

Your turn

Your turn · student questions

Piecewise Functions · Choose the Interval First

1. Piece 1

2. Piece 2

3. Piece 3

4. Endpoints

5. Final graph

1. First rule

2. Second rule

3. Boundary x=2

4. Final graph

Your turn

Your turn · student questions

Absolute Value → Piecewise

1. For x≥0

2. For x<0

3. Write one function

4. Final graph

1. Breaking point

2. Left of 3

3. Right of 3

4. Final graph

1. Breaking point

2. x≤1

3. x>1

4. Final graph

1. Factor inside

2. Breaking points

3. Sign intervals

4. Apply the sign

5. Final graph

Your turn

Your turn · student questions

Graph → Equation

1. Start with vertex form

2. Use the second point

3. Solve

4. Equation

Your turn

Your turn · student questions

Greatest Integer Functions

1. Step length

2. Boundary pattern

3. Build intervals

4. Domain and range

5. Final graph

1. Coefficient of x

2. Step length

3. Build from boundaries

4. Final graph

1. Step length

2. Direction

3. Respect the domain

4. Final graph

Your turn

Your turn · student questions

Vertical Line Test

Your turn

Your turn · student questions

Does an Equation Define y as a Function of x?

1. Solve for y

2. Decide

1. Isolate y²

2. Solve for y

3. Decide

1. Factor y

2. Solve

3. Decide