Quadratic Functions Revisited

You built parabolas all through Grade 10; today at the boards they came back wearing function notation and answering harder questions. The parabola itself has not changed. What changes this year is fluency: choosing the form of a quadratic to match the question being asked.

Three forms, three superpowers

Form	Looks like	Reveals instantly
Standard	$f(x)=a{x}^2+bx+c$	the $y$-intercept, $c$
Vertex	$f(x)=a(x-h{)}^2+k$	the max or min, at $(h,k)$
Factored	$f(x)=a(x-r)(x-s)$	the zeros, $r$ and $s$

Factored form also explains families: every quadratic with zeros $-1$ and $3$ is $f(x)=a(x+1)(x-3)$ for some $a$ — same anchors, different stretch. One extra point pins down the member. Through $(1,8)$: substituting gives $a(2)(-2)=8$, so $a=-2$ and $f(x)=-2(x+1)(x-3)$.

Max and min without a graph

Completing the square converts standard form to vertex form even when $a\ne 1$ — factor $a$ out of the $x$-terms first:

$\begin{array}{rl}f(x)& =2x^2-12x+7=2(x^2-6x)+7=2(x-3{)}^2-18+7\\ & =2(x-3{)}^2-11\end{array}$

Minimum value $-11$, at $x=3$. A slicker route when it applies: partial factoring. Write $f(x)=3x^2-6x+5$ as $3x(x-2)+5$; the function takes the value 5 at both $x=0$ and $x=2$, symmetry puts the vertex halfway between at $x=1$, and $f(1)=2$. In an application, this vertex is the answer — the maximum profit, the peak height — so translate back into a sentence when you get there.

Counting zeros before finding them

The discriminant $b^2-4ac$ announces how many $x$-intercepts a quadratic has before you hunt for them: positive means two, zero means exactly one, negative means none. Vertex form tells the same story geometrically — a vertex below the axis on a parabola opening up must cross twice. Predicting the count first, then verifying, is a favourite move in Graph Talks, and it is Checking Your Own Work built into the solving itself: if you predicted two zeros and found one, something is asking to be re-examined.

These moves get exercised whenever quadratics appear this semester — Function Notation Practice and Transformations Practice both lean on them.

Curriculum connection

A2.1

determine the number of zeros (i.e., x-intercepts) of a quadratic function, using a variety of strategies (e.g., inspecting graphs; factoring; calculating the discriminant) Sample problem: Investigate, using graphing technology and algebraic techniques, the transformations that affect the number of zeros for a given quadratic function.

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A2.2

determine the maximum or minimum value of a quadratic function whose equation is given in the form $f(x)=a{x}^2+bx+c$, using an algebraic method (e.g., completing the square; factoring to determine the zeros and averaging the zeros) Sample problem: Explain how partially factoring $f(x)=3x^2-6x+5$ into the form $f(x)=3x(x-2)+5$ helps you determine the minimum of the function.

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A2.3

solve problems involving quadratic functions arising from real-world applications and represented using function notation Sample problem: The profit, $P(x)$, of a video company, in thousands of dollars, is given by $P(x)=-5x^2+550x-5000$, where $x$ is the amount spent on advertising, in thousands of dollars. Determine the maximum profit that the company can make, and the amounts spent on advertising that will result in a profit and that will result in a profit of at least $4 000 000.

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A2.4

determine, through investigation, the transformational relationship among the family of quadratic functions that have the same zeros, and determine the algebraic representation of a quadratic function, given the real roots of the corresponding quadratic equation and a point on the function Sample problem: Determine the equation of the quadratic function that passes through $(2,5)$ if the roots of the corresponding quadratic equation are $1+\sqrt{5}$ and $1-\sqrt{5}$.

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