Shifting the parabola

Algebra, grade 9 Lesson 20 of 26Learning

Shifting the parabola

Shifting the parabola $\displaystyle {y}={ax^{2}}$ gives the parabolas $\displaystyle {y}={a(x-b)^{2}+k}$ . The shifts are those of a function’s graph:

adding $\displaystyle k$ moves the parabola up or down;
$\displaystyle x-b$ in place of $\displaystyle x$ moves it right or left.

Vertical shift:

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

The parabola $\displaystyle {y}={x^{2}-4}$ (dashed) is $\displaystyle {y}={x^{2}}$ moved 4 down. Its vertex is $\displaystyle (0,-4)$ .

\displaystyle y=x^{2}

\displaystyle y=x^{2}-4

\displaystyle A=(-2,0)

\displaystyle B=(2,0)

\displaystyle V=(0,-4)

Its intersection points with the $\displaystyle x$ -axis are where $\displaystyle {y}={0}$ : $\displaystyle {x^{2}-4}={0}$ , so $\displaystyle {x^{2}}={4}$ , and $\displaystyle {x}={2}$ or $\displaystyle {x}={-2}$ – the points $\displaystyle A(-2,0)$ and $\displaystyle B(2,0)$ .

Horizontal shift:

\displaystyle {y}={a(x-b)^{2}}

\displaystyle {y}={(x+3)^{2}}

is a shift 3 to the left: its vertex is

\displaystyle (-3,0)

The parabola $\displaystyle {y}={(x-3)^{2}}$ (dashed) is $\displaystyle {y}={x^{2}}$ moved 3 to the right. Its vertex is $\displaystyle (3,0)$ , and its axis of symmetry is the line $\displaystyle {x}={3}$ .

\displaystyle y=x^{2}

\displaystyle y=(x-3)^{2}

\displaystyle V=(3,0)

The vertex form:

\displaystyle {y}={a(x-b)^{2}+k}

The parabola $\displaystyle {y}={a(x-b)^{2}+k}$ is $\displaystyle {y}={ax^{2}}$ moved $\displaystyle b$ to the right and $\displaystyle k$ up.

Its vertex is $\displaystyle (b,k)$ , and its axis of symmetry is the line $\displaystyle {x}={b}$ .
$\displaystyle a>0$ : the vertex is a minimum point; $\displaystyle a<0$ : a maximum point.

For example, $\displaystyle {y}={(x-1)^{2}-4}$ : vertex $\displaystyle (1,-4)$ , a minimum. $\displaystyle {y}={-(x+2)^{2}+1}$ : vertex $\displaystyle (-2,1)$ , a maximum.

The properties of

\displaystyle {y}={(x-1)^{2}-4}

\displaystyle {(x-1)^{2}}={4}

has two solutions:

\displaystyle x-1

\displaystyle 2

\displaystyle -2

Vertex $\displaystyle V(1,-4)$ , a minimum point; the minimum value is $\displaystyle -4$ .
Decreasing for $\displaystyle x<1$ , increasing for $\displaystyle x>1$ .
The $\displaystyle y$ -axis, $\displaystyle {x}={0}$ : $\displaystyle {y}={(0-1)^{2}-4}={-3}$ – the point $\displaystyle C(0,-3)$ .
The $\displaystyle x$ -axis, $\displaystyle {y}={0}$ : $\displaystyle {(x-1)^{2}-4}={0}$ , so $\displaystyle {(x-1)^{2}}={4}$ ; $\displaystyle {x-1}={2}$ or $\displaystyle {x-1}={-2}$ ; $\displaystyle {x}={3}$ or $\displaystyle {x}={-1}$ – the points $\displaystyle A(-1,0)$ and $\displaystyle B(3,0)$ .
Positive for $\displaystyle x<-1$ and for $\displaystyle x>3$ ; negative for $\displaystyle -1<x<3$ .

\displaystyle y=(x-1)^{2}-4

\displaystyle V=(1,-4)

\displaystyle A=(-1,0)

\displaystyle B=(3,0)

\displaystyle C=(0,-3)

The properties of

\displaystyle {y}={-(x+2)^{2}+1}

Vertex $\displaystyle V(-2,1)$ , a maximum point; the maximum value is 1.
Increasing for $\displaystyle x<-2$ , decreasing for $\displaystyle x>-2$ .
The $\displaystyle y$ -axis: $\displaystyle {y}={-(0+2)^{2}+1}={-3}$ – the point $\displaystyle C(0,-3)$ .
The $\displaystyle x$ -axis: $\displaystyle {(x+2)^{2}}={1}$ ; $\displaystyle {x+2}={1}$ or $\displaystyle {x+2}={-1}$ ; $\displaystyle {x}={-1}$ or $\displaystyle {x}={-3}$ – the points $\displaystyle A(-3,0)$ , $\displaystyle B(-1,0)$ .
Positive for $\displaystyle -3<x<-1$ ; negative for $\displaystyle x<-3$ and for $\displaystyle x>-1$ .

\displaystyle y=-(x+2)^{2}+1

\displaystyle V=(-2,1)

\displaystyle A=(-3,0)

\displaystyle B=(-1,0)

\displaystyle C=(0,-3)

A parabola above the

\displaystyle x

-axis

The parabola $\displaystyle {y}={x^{2}+2}$ has its minimum point $\displaystyle (0,2)$ above the $\displaystyle x$ -axis, so it does not meet the $\displaystyle x$ -axis: the equation $\displaystyle {x^{2}+2}={0}$ , that is $\displaystyle {x^{2}}={-2}$ , has no solution. The same goes for a parabola with a maximum point below the $\displaystyle x$ -axis.

Exercise 1

For each parabola write: the vertex – minimum or maximum, the axis of symmetry, the increasing and decreasing domains, the intersection points with the axes, the positive and negative domains.

$\displaystyle {y}={(x-2)^{2}-9}$

/ – 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

$\displaystyle {y}={-(x+1)^{2}+4}$

/ – 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

$\displaystyle {y}={2(x-3)^{2}}$

/ – 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

$\displaystyle {y}={-x^{2}-5}$

Exercise 2

Write the equation of a parabola whose vertex, $\displaystyle (4,-1)$ , is a minimum point.

Exercise 3

Write the equation of each parabola:

$\displaystyle {y}={x^{2}}$ moved 5 to the left and 2 up;
$\displaystyle {y}={-x^{2}}$ moved 3 to the right.

What is the maximum point of the second parabola?

\displaystyle y=x^{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 $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$