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Core notes

A quadratic function links an algebraic expression to a parabola. Completing the square reveals the turning point, the discriminant controls whether the graph meets the horizontal axis, and a model is useful only when its variables and domain match the situation.

Maximum and minimum by completing the square

Write $ax^2+bx+c$ as $a(x-h)^2+k$ . Because a square is non-negative, the sign of $a$ shows whether $k$ is a minimum or maximum value. The turning point is $(h,k)$ , and the axis of symmetry is $x=h$ .

When a quadratic is always positive or negative

For $y=ax^2+bx+c$ to be always positive, the parabola must open upward and never touch the horizontal axis: $a>0$ and $b^2-4ac<0$ . For it to be always negative, require $a<0$ and $b^2-4ac<0$ . If the discriminant is zero, the function reaches zero and is only non-negative or non-positive.

Quadratic models

A quadratic may model height, area, revenue, or another quantity with one turning point. Define the variable, translate the context into a function, and restrict the domain to meaningful values. Interpret the calculated extremum with units and reject algebraic values that do not fit the situation.

Relationship

Use

ax^2+bx+c=a(x-h)^2+k

Expose the turning point and extreme value.

x_{\text{axis}}=-\frac{b}{2a}

Locate the axis of symmetry.

\Delta=b^2-4ac

Test how many real horizontal-axis intersections exist.

Worked examples

Example 1: Find the minimum value of $2x^2-12x+23$ and the value of $x$ where it occurs.

Factor the leading coefficient from the first two terms: $2(x^2-6x)+23$ .
Complete the square: $2[(x-3)^2-9]+23=2(x-3)^2+5$ .
Since $2(x-3)^2\geq0$ , the minimum is reached when $x=3$ .

Answer: The minimum value is $5$ , occurring at $x=3$ .

Example 2: Find the values of $k$ for which $x^2+4x+k$ is always positive.

The leading coefficient is positive.
Require the discriminant to be negative: $4^2-4(1)(k)<0$ .
Solve $16-4k<0$ , giving $k>4$ .

Answer: The expression is always positive when $k>4$ .

Exam traps and retrieval check

Avoid these traps

Reading the turning point from an incomplete square with the wrong sign.
Using $\Delta\leq0$ for always positive and forgetting that $\Delta=0$ allows the value zero.
Reporting a model solution without checking its domain or units.

Check from memory

What does the sign of

a

tell you?

It tells whether the parabola opens upward or downward and therefore whether the turning point is a minimum or maximum.

What conditions make

ax^2+bx+c

always negative?

$a<0$ and $b^2-4ac<0$ .

Why should a quadratic model have a stated domain?

The algebraic function may allow inputs that have no meaning in the original situation.

Core notes

Maximum and minimum by completing the square

When a quadratic is always positive or negative

Quadratic models

Relationship

Use

ax^2+bx+c=a(x-h)^2+k

Expose the turning point and extreme value.

x_{\text{axis}}=-\frac{b}{2a}

Locate the axis of symmetry.

\Delta=b^2-4ac

Test how many real horizontal-axis intersections exist.

Worked examples

Example 1: Find the minimum value of $2x^2-12x+23$ and the value of $x$ where it occurs.

Factor the leading coefficient from the first two terms: $2(x^2-6x)+23$ .
Complete the square: $2[(x-3)^2-9]+23=2(x-3)^2+5$ .
Since $2(x-3)^2\geq0$ , the minimum is reached when $x=3$ .

Answer: The minimum value is $5$ , occurring at $x=3$ .

Example 2: Find the values of $k$ for which $x^2+4x+k$ is always positive.

The leading coefficient is positive.
Require the discriminant to be negative: $4^2-4(1)(k)<0$ .
Solve $16-4k<0$ , giving $k>4$ .

Answer: The expression is always positive when $k>4$ .

Exam traps and retrieval check

Avoid these traps

Reading the turning point from an incomplete square with the wrong sign.
Using $\Delta\leq0$ for always positive and forgetting that $\Delta=0$ allows the value zero.
Reporting a model solution without checking its domain or units.

Check from memory

What does the sign of

a

tell you?

It tells whether the parabola opens upward or downward and therefore whether the turning point is a minimum or maximum.

What conditions make

ax^2+bx+c

always negative?

$a<0$ and $b^2-4ac<0$ .

Why should a quadratic model have a stated domain?

The algebraic function may allow inputs that have no meaning in the original situation.

A1: Quadratic functions

Core notes

Maximum and minimum by completing the square

When a quadratic is always positive or negative

Quadratic models

Formulae and methods

Worked examples

Example 1: Find the minimum value of $2x^2-12x+23$ and the value of $x$ where it occurs.

Example 2: Find the values of $k$ for which $x^2+4x+k$ is always positive.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

A1: Quadratic functions

Core notes

Maximum and minimum by completing the square

When a quadratic is always positive or negative

Quadratic models

Formulae and methods

Worked examples

Example 1: Find the minimum value of $2x^2-12x+23$ and the value of $x$ where it occurs.

Example 2: Find the values of $k$ for which $x^2+4x+k$ is always positive.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

Relationship	Use
$ax^2+bx+c=a(x-h)^2+k$	Expose the turning point and extreme value.
$x_{\text{axis}}=-\frac{b}{2a}$	Locate the axis of symmetry.
$\Delta=b^2-4ac$	Test how many real horizontal-axis intersections exist.

Core notes

Maximum and minimum by completing the square

When a quadratic is always positive or negative

Quadratic models

Formulae and methods

Worked examples

Example 1: Find the minimum value of 2x2−12x+232x^2-12x+232x2−12x+23 and the value of xxx where it occurs.

Example 2: Find the values of kkk for which x2+4x+kx^2+4x+kx2+4x+k is always positive.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

Core notes

Maximum and minimum by completing the square

When a quadratic is always positive or negative

Quadratic models

Formulae and methods

Worked examples

Example 1: Find the minimum value of 2x2−12x+232x^2-12x+232x2−12x+23 and the value of xxx where it occurs.

Example 2: Find the values of kkk for which x2+4x+kx^2+4x+kx2+4x+k is always positive.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

Example 1: Find the minimum value of $2x^2-12x+23$ and the value of $x$ where it occurs.

Example 2: Find the values of $k$ for which $x^2+4x+k$ is always positive.

Example 1: Find the minimum value of $2x^2-12x+23$ and the value of $x$ where it occurs.

Example 2: Find the values of $k$ for which $x^2+4x+k$ is always positive.