The function y = ax²

Algebra, grade 9 Lesson 19 of 26Learning

The function y = ax²

A quadratic function has $\displaystyle x^{2}$ as its highest power. The simplest one is $\displaystyle {y}={x^{2}}$ , and its graph is a curve called a parabola.

A table and a graph of

\displaystyle {y}={x^{2}}

A negative number squared is positive:

\displaystyle {(-3)^{2}}={9}

$\displaystyle x$	$\displaystyle -3$	$\displaystyle -2$	$\displaystyle -1$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 3$
$\displaystyle {y}={x^{2}}$	$\displaystyle 9$	$\displaystyle 4$	$\displaystyle 1$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 4$	$\displaystyle 9$

The points of the table, joined by a smooth curve:

\displaystyle y=x^{2}

\displaystyle (-3,9)

\displaystyle (-2,4)

\displaystyle (-1,1)

\displaystyle (0,0)

\displaystyle (1,1)

\displaystyle (2,4)

\displaystyle (3,9)

The vertex and the axis of symmetry

The vertex of a parabola is its lowest point (or its highest). The vertex of $\displaystyle {y}={x^{2}}$ is $\displaystyle (0,0)$ .
The axis of symmetry is the vertical line through the vertex: the two halves of the parabola are mirror images in it. For $\displaystyle {y}={x^{2}}$ it is the $\displaystyle y$ -axis, $\displaystyle {x}={0}$ – since $\displaystyle x$ and $\displaystyle -x$ give the same $\displaystyle y$ : $\displaystyle {(-3)^{2}}={3^{2}}={9}$ .

The sign of

\displaystyle a

: minimum or maximum

\displaystyle {y}={-x^{2}}

the square comes first, then the minus: at

\displaystyle {x}={3}

\displaystyle {y}={-9}

In $\displaystyle {y}={ax^{2}}$ ( $\displaystyle a\neq0$ ), the sign of $\displaystyle a$ tells which way the parabola opens:

$\displaystyle a>0$ : it opens upward, and the vertex is a minimum point – $\displaystyle {y}={x^{2}}$ (solid), $\displaystyle {y}={2x^{2}}$ .
$\displaystyle a<0$ : it opens downward, and the vertex is a maximum point – $\displaystyle {y}={-x^{2}}$ (dashed).

\displaystyle y=x^{2}

\displaystyle y=-x^{2}

The properties of

\displaystyle {y}={ax^{2}}

	$\displaystyle a>0$	$\displaystyle a<0$
vertex	$\displaystyle (0,0)$ – a minimum point	$\displaystyle (0,0)$ – a maximum point
increasing	$\displaystyle x>0$	$\displaystyle x<0$
decreasing	$\displaystyle x<0$	$\displaystyle x>0$
intersection with the axes	only $\displaystyle (0,0)$	only $\displaystyle (0,0)$
positive	every $\displaystyle x\neq0$	never
negative	never	every $\displaystyle x\neq0$

Exercise 1

The function $\displaystyle {y}={2x^{2}}$ :

Complete a table for $\displaystyle {x}={-3,-2,-1,0,1,2,3}$ .
Sketch its graph.
Write its properties: vertex, minimum or maximum, increasing and decreasing domains.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

The function is $\displaystyle {y}={-2x^{2}}$ .
Calculate its value at $\displaystyle {x}={3}$ and at $\displaystyle {x}={-2}$ .
Write its properties: vertex, minimum or maximum, increasing and decreasing domains, positive and negative domains.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Which of these points are on the graph of $\displaystyle {y}={-x^{2}}$ : $\displaystyle (3,-9)$ , $\displaystyle (-2,4)$ , $\displaystyle (-1,-1)$ , $\displaystyle (0,0)$ ?
Which of these functions have a maximum point: $\displaystyle {y}={5x^{2}}$ , $\displaystyle {y}={-0.5x^{2}}$ , $\displaystyle {y}={0.1x^{2}}$ , $\displaystyle {y}={-x^{2}}$ ?

Formulas: algebra, grade 9

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The laws of powers

Multiplying powers with the same base
$\displaystyle a^{m}\cdot a^{n}$ $\displaystyle =$ $\displaystyle a^{m+n}$
Dividing powers with the same base
$\displaystyle a^{m}:a^{n}$ $\displaystyle =$ $\displaystyle a^{m-n}$ $\displaystyle a\neq0$
A power of a power
$\displaystyle \left(a^{m}\right)^{n}$ $\displaystyle =$ $\displaystyle a^{m\cdot n}$
A power of a product
$\displaystyle (a\cdot b)^{n}$ $\displaystyle =$ $\displaystyle a^{n}\cdot b^{n}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{n}$ $\displaystyle =$ $\displaystyle \frac{a^{n}}{b^{n}}$ $\displaystyle b\neq0$
The power zero
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$

The laws of square roots

The root of a product
$\displaystyle \sqrt{a\cdot b}$ $\displaystyle =$ $\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle a\ge0,\ b\ge0$
The root of a quotient
$\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle =$ $\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle a\ge0,\ b>0$
The square of a root
$\displaystyle \left(\sqrt{a}\right)^{2}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$
The root of a square
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The extended distributive law and the short multiplication formulas

The extended distributive law
$\displaystyle (a+b)(c+d)$ $\displaystyle =$ $\displaystyle ac+ad+bc+bd$
The square of a sum
$\displaystyle (a+b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}+2ab+b^{2}$
The square of a difference
$\displaystyle (a-b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}-2ab+b^{2}$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^{2}-b^{2}$

The parabola

Standard form
$\displaystyle y=ax^{2}+bx+c$
Vertex form – the vertex is $\displaystyle (b,k)$
$\displaystyle y=a(x-b)^{2}+k$
Product form – it meets the $\displaystyle x$ -axis at $\displaystyle x_1$ and $\displaystyle x_2$
$\displaystyle y=a(x-x_1)(x-x_2)$
The vertex’s $\displaystyle x$ , in standard form
$\displaystyle x=-\frac{b}{2a}$

The quadratic equation

The roots of $\displaystyle {ax^{2}+bx+c}={0}$ , when $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$