Stretching, shrinking and the absolute value

Algebra and calculus, grade 10 (4 units) Lesson 7 of 20Learning

Stretching, shrinking and the absolute value

Stretching and shrinking

Multiplying a function by a number multiplies every value by it, and the graph is stretched or shrunk along the $\displaystyle y$ -axis:

\displaystyle y=a\cdot f(x)

$\displaystyle |a|>1$ – a stretch: the graph moves away from the $\displaystyle x$ -axis.
$\displaystyle |a|<1$ – a shrink: the graph moves closer to the $\displaystyle x$ -axis.
$\displaystyle a<0$ – a reflection in the $\displaystyle x$ -axis as well.

On the board: $\displaystyle {f(x)}={x^2}$ , $\displaystyle {g(x)}={2x^2}$ and $\displaystyle {h(x)}={\frac{1}{2}x^2}$ (the dashed ones).

\displaystyle f(x)=x^{2}

\displaystyle g(x)=2x^{2}

\displaystyle h(x)=\frac{1}{2}x^{2}

\displaystyle (1,1)

\displaystyle (1,2)

The point $\displaystyle (1,1)$ of $\displaystyle f$ moves to $\displaystyle (1,2)$ on $\displaystyle g$ : the $\displaystyle x$ stays, the $\displaystyle y$ is doubled.
Every point $\displaystyle (x_0,y_0)$ of $\displaystyle f$ moves to $\displaystyle (x_0,\ a\cdot y_0)$ .
The points where the graph meets the $\displaystyle x$ -axis do not move: $\displaystyle 0$ times any number is $\displaystyle 0$ .

The absolute value

The absolute value of a number is its distance from 0 on the number line: $\displaystyle {|5|}={5}$ , $\displaystyle {|-5|}={5}$ , $\displaystyle {|0|}={0}$ .

\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}

On the board: $\displaystyle {y}={|x|}$ .

\displaystyle y=|x|

To the right of 0 the graph is $\displaystyle {y}={x}$ ; to the left of 0 it is $\displaystyle {y}={-x}$ .
An absolute value is never negative, so the whole graph is above the $\displaystyle x$ -axis or on it.

The graph of

\displaystyle {y}={|f(x)|}

In the graph of $\displaystyle {y}={|f(x)|}$ everything above the $\displaystyle x$ -axis stays, and everything below it is reflected above it.

On the board: $\displaystyle {f(x)}={x^2-4}$ (the dashed one) and $\displaystyle {g(x)}={|x^2-4|}$ .

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

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

\displaystyle (0,4)

Between $\displaystyle {x}={-2}$ and $\displaystyle {x}={2}$ the values of $\displaystyle f$ are negative, and there the graph of $\displaystyle g$ is their reflection: $\displaystyle {g(x)}={-f(x)}$ .
The minimum point of $\displaystyle f$ , $\displaystyle (0,-4)$ , moves to $\displaystyle (0,4)$ : a maximum point of $\displaystyle g$ .
At the points $\displaystyle (-2,0)$ and $\displaystyle (2,0)$ the value of $\displaystyle g$ is 0, the least it has: these are the minimum points of $\displaystyle g$ .

Several transformations together

$\displaystyle {g(x)}={2(x-1)^2-3}$ .
The example below gets the graph of $\displaystyle g$ from the graph of $\displaystyle {f(x)}={x^2}$ in three steps.

A stretch by 2: $\displaystyle {y}={2x^2}$ .
A shift of one unit to the right: $\displaystyle {y}={2(x-1)^2}$ .
A shift of 3 units down: $\displaystyle {y}={2(x-1)^2-3}$ .

\displaystyle f(x)=x^{2}

\displaystyle g(x)=2(x-1)^{2}-3

\displaystyle V=(1,-3)

The vertex $\displaystyle (0,0)$ does not move in the stretch, and the two shifts take it to $\displaystyle (1,-3)$ .
The order matters: shifting 3 down first and stretching afterwards gives $\displaystyle {2(x^2-3)}={2x^2-6}$ – another function.

Exercise 1

On the board: $\displaystyle {f(x)}={x^2}$ , and two graphs whose formulas are hidden.
Each of them is $\displaystyle a\cdot f(x)$ , for another $\displaystyle a$ .
Write the formula of each graph in a row of its own.

\displaystyle f(x)=x^{2}

\displaystyle g(x)=\,?

\displaystyle h(x)=\,?

Exercise 2

$\displaystyle {f(x)}={x^2-2x-3}$ .
$\displaystyle {g(x)}={|x^2-2x-3|}$ .
The minimum point of $\displaystyle f$ is $\displaystyle (1,-4)$ .

Mark the points where the graph of $\displaystyle g$ meets the $\displaystyle x$ -axis.

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

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

What are the minimum points of $\displaystyle g$ ?
What is the maximum point of $\displaystyle g$ between the points where it meets the axis?

Exercise 3

$\displaystyle {f(x)}={\sqrt{x}}$ .
The graph of $\displaystyle g$ is the graph of $\displaystyle f$ , stretched by 2 and shifted one unit to the right.

Write the formula of $\displaystyle g$ on the board.

\displaystyle f(x)=\sqrt{x}

\displaystyle g(x)=\,?

What is the domain of $\displaystyle g$ ?
At which point does its graph start?

Exercise 4

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

How do you get the graph of $\displaystyle g$ from the graph of $\displaystyle {y}={x^2}$ ?
Write the steps in order.

Mark the extremum point of $\displaystyle g$ and the points where its graph meets the $\displaystyle x$ -axis.

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

Exercise 5

$\displaystyle f(|x|)$ means: put $\displaystyle |x|$ in $\displaystyle f$ in place of $\displaystyle x$ . For example, if $\displaystyle {f(x)}={x-1}$ , then $\displaystyle {f(|x|)}={|x|-1}$ .
Is the claim true for every function $\displaystyle f$ ?
If so – explain why. If not – give a counterexample.

The function $\displaystyle |f(x)|$ is an even function.

The function $\displaystyle f(|x|)$ is an even function.

The graph of $\displaystyle {y}={|f(x)|}$ passes through the origin.

No value of the function $\displaystyle |f(x)|$ is negative.

Formulas: algebra and calculus, grade 10 (4 units)

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Algebraic technique

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
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^2-b^2$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$
The roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
The product of roots
$\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle =$ $\displaystyle \sqrt{ab}$ $\displaystyle a\geq0,\ b\geq0$
The quotient of roots
$\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle =$ $\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle a\geq0,\ b>0$

The properties of a function

An even function – its graph symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – its graph symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The $\displaystyle x$ -coordinate of the vertex of $\displaystyle {y}={ax^2+bx+c}$
$\displaystyle x=\frac{-b}{2a}$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transforming a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units to the right ( $\displaystyle p<0$ – to the left)
$\displaystyle y=f(x-p)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A stretch ( $\displaystyle |a|>1$ ) or a shrink ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$

The derivative and the rules of differentiation

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the chord
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The derivative of a product
$\displaystyle (f(x)\cdot g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a composite function
$\displaystyle (f(g(x)))'$ $\displaystyle =$ $\displaystyle f'(g(x))\cdot g'(x)$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

The derivative of a square root function

The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$