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Quadratics · eight-question check

Read the idea, work independently, then explain what changed.

TOPIC 01

Quadratics

Try two tasks per skill independently, then revisit the linked skill.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Which is the axis?

y=(x+3)^2-4

All inputs are real unless a smaller domain is stated.

Skills and prerequisite lessons

Scope: JM01 / JM02

x=−3
x=3
x=−4
x=4

Working and explanation

BUILD THE REASONING

Hint 1

Locate the zero square.

Hint 2

Solve x+3=0.

Worked solution

Set the base to zero.
$x+3=0$
The height constant does not change the axis.
$x=-3$

A: x=−3.

Checks and common pitfalls: Read the coordinate, not the visible plus sign.

Think first. Reveal a hint when the class is ready.

02 / Foundation#Your turn

Which statement holds?

y=-2(x-1)^2+3

All inputs are real unless a smaller domain is stated.

Skills and prerequisite lessons

Scope: JM01 / JM02

Minimum 3
Maximum 3
Maximum −2
No extremum

Working and explanation

BUILD THE REASONING

Hint 1

Check the square coefficient.

Hint 2

A negative square term is nonpositive.

Worked solution

Bound the changing term.
$-2(x-1)^2\le0$
Equality is feasible at x=1.
$y\le3$

B: maximum 3.

Checks and common pitfalls: Opening direction decides the type of extremum.

Think first. Reveal a hint when the class is ready.

03 / Foundation#Your turn

What does Δ=0 mean for a genuine quadratic?

a\ne0

a≠0.

Skills and prerequisite lessons

Scope: JM01 / JM02

No real roots
One distinct real root
Two distinct real roots
Infinitely many

Working and explanation

BUILD THE REASONING

Hint 1

Use the square equation.

Hint 2

Zero square has one base value.

Worked solution

Set the squared base to zero.
$2ax+b=0$
There is one distinct input.
$x=-b/(2a)$

B: one distinct real root.

Checks and common pitfalls: Repeated multiplicity is not a second input.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Choose the complete solution set.

x(x-3)=0

All inputs are real unless a smaller domain is stated.

Skills and prerequisite lessons

Scope: JM01 / JM02

{3}
{0}
{0,3}
{0,−3}

Working and explanation

BUILD THE REASONING

Hint 1

Do not divide by x yet.

Hint 2

Either factor can be zero.

Worked solution

Treat the first factor.
$x=0$
Treat the second factor.
$x=3$

C: {0,3}.

Checks and common pitfalls: Division by x without a zero case loses a valid root.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Which is the solution?

-(x-1)(x-3)>0

All inputs are real unless a smaller domain is stated.

Skills and prerequisite lessons

Scope: JM01 / JM02

Outside [1,3]
(1,3)
[1,3]
{1,3}

Working and explanation

BUILD THE REASONING

Hint 1

Account for the leading minus.

Hint 2

The product before negation must be negative.

Worked solution

Reverse the target sign.
$(x-1)(x-3)<0$
Choose between the roots.
$1<x<3$

B: (1,3).

Checks and common pitfalls: A memorised outside rule fails when the leading sign changes.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Solve the strict inequality.

(x-2)^2<0

All inputs are real unless a smaller domain is stated.

Skills and prerequisite lessons

Scope: JM01 / JM02

{2}
x<2
Empty set
All real x

Working and explanation

BUILD THE REASONING

Hint 1

Use nonnegativity of a real square.

Hint 2

Even x=2 gives zero, not a negative value.

Worked solution

Bound every output.
$(x-2)^2\ge0$
No real input meets the strict bound.
$\varnothing$

C: empty set.

Checks and common pitfalls: A repeated root cannot make a square negative.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Where is the minimum attained?

f=(x-3)^2,\quad x\in[0,1]

All inputs are real unless a smaller domain is stated.

Skills and prerequisite lessons

Scope: JM01 / JM02

x=3
x=1
x=0
No minimum

Working and explanation

BUILD THE REASONING

Hint 1

Mark the permitted interval.

Hint 2

Use the permitted input nearest 3.

Worked solution

Compare endpoint distances.
$|1-3|=2<3=|0-3|$
The vertex is not in the domain.
$x=1$

B: x=1.

Checks and common pitfalls: An unrestricted vertex rule cannot override the domain.

Think first. Reveal a hint when the class is ready.

08 / Foundation#Your turn

Which statement is correct?

f=x^2,\quad x\in(0,1)

0<x<1.

Skills and prerequisite lessons

Scope: JM01 / JM02

Minimum 0
Maximum 1
Neither exists
Unbounded

Working and explanation

BUILD THE REASONING

Hint 1

Can x=0 or x=1 be used?

Hint 2

Outputs approach both bounds but do not reach them.

Worked solution

Keep strict endpoint restrictions.
$0<f<1$
No output attains either bound.
$\inf f=0,\quad\sup f=1$

C: neither minimum nor maximum exists.

Checks and common pitfalls: A bounded function need not attain extrema on an open domain.

Think first. Reveal a hint when the class is ready.