Shifts, stretches and reflections

Calculus, grade 10 (5 units) Lesson 5 of 26Learning

Shifts, stretches and reflections

A shift

You met the shifts of the parabola in grade 9. They work on the graph of any function:

\displaystyle y=f(x)+k

\displaystyle y=f(x-p)

$\displaystyle {y}={f(x)+k}$ : every point goes up $\displaystyle k$ units ( $\displaystyle k<0$ – down).
$\displaystyle {y}={f(x-p)}$ : every point moves $\displaystyle p$ units right ( $\displaystyle p<0$ – left). The value $\displaystyle f$ took at $\displaystyle x$ , the new function takes at $\displaystyle x+p$ .

On the board below, $\displaystyle {f(x)}={x^2}$ and $\displaystyle {g(x)}={(x-2)^2+1}$ : the vertex $\displaystyle (0,0)$ moves 2 right and 1 up, to $\displaystyle (2,1)$ .

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

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

\displaystyle V=(2,1)

A vertical stretch

\displaystyle y=a\cdot f(x)

In $\displaystyle {y}={a\cdot f(x)}$ every value of $\displaystyle y$ is multiplied by $\displaystyle a$ , and $\displaystyle x$ does not change:

$\displaystyle a>1$ : the graph is stretched away from the $\displaystyle x$ -axis ( $\displaystyle 2x^2$ is narrower than $\displaystyle x^2$ ). $\displaystyle 0<a<1$ : the graph is squeezed towards the axis.
The zeros stay where they are: $\displaystyle {a\cdot0}={0}$ .
An extremum point stays at the same $\displaystyle x$ , and its $\displaystyle y$ -coordinate is multiplied by $\displaystyle a$ .

A horizontal squeeze

\displaystyle y=f(ax)

In $\displaystyle {y}={f(ax)}$ ( $\displaystyle a>0$ ) the new function takes at $\displaystyle x$ the value $\displaystyle f$ takes at $\displaystyle ax$ . So every point $\displaystyle (x,y)$ of $\displaystyle f$ moves to $\displaystyle \left(\frac{x}{a},y\right)$ :

$\displaystyle a>1$ : the graph is squeezed towards the $\displaystyle y$ -axis. $\displaystyle 0<a<1$ : it is stretched away from it.
The zeros are divided by $\displaystyle a$ , and the values of $\displaystyle y$ do not change.

On the board below, $\displaystyle {f(x)}={x^2-4}$ , whose zeros are $\displaystyle \pm2$ , and $\displaystyle {f(2x)}={(2x)^2-4}={4x^2-4}$ , whose zeros are $\displaystyle \pm1$ .

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

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

A reflection

\displaystyle y=-f(x)

\displaystyle y=f(-x)

$\displaystyle {y}={-f(x)}$ : a reflection in the $\displaystyle x$ -axis. Every point $\displaystyle (x,y)$ moves to $\displaystyle (x,-y)$ ; a maximum becomes a minimum.
$\displaystyle {y}={f(-x)}$ : a reflection in the $\displaystyle y$ -axis. Every point $\displaystyle (x,y)$ moves to $\displaystyle (-x,y)$ .
An even function does not change when reflected in the $\displaystyle y$ -axis: $\displaystyle {f(-x)}={f(x)}$ .

What happens to a point

The table below shows where a point $\displaystyle (x,y)$ of the graph of $\displaystyle f$ moves to:

The function	The point moves to
$\displaystyle f(x)+k$	$\displaystyle (x,y+k)$
$\displaystyle f(x-p)$	$\displaystyle (x+p,y)$
$\displaystyle a\cdot f(x)$	$\displaystyle (x,ay)$
$\displaystyle f(ax)$	$\displaystyle \left(\frac{x}{a},y\right)$
$\displaystyle -f(x)$	$\displaystyle (x,-y)$
$\displaystyle f(-x)$	$\displaystyle (-x,y)$

The zeros and the extremum points move this way too. Shifting and changing the shape do not change the number of extremum points.

The family

\displaystyle a(x-p)^n+k

Every graph of the form $\displaystyle {y}={a(x-p)^n+k}$ is built from the graph of $\displaystyle {y}={x^n}$ : a stretch by $\displaystyle a$ (and a reflection when $\displaystyle a<0$ ), a shift of $\displaystyle p$ to the right and $\displaystyle k$ up. The point $\displaystyle (0,0)$ of $\displaystyle x^n$ moves to $\displaystyle (p,k)$ .

The example below builds $\displaystyle {y}={-2(x-1)^3+4}$ from $\displaystyle {y}={x^3}$ (dashed on the board):

$\displaystyle {a}={-2}$ : a stretch by 2 and a reflection in the $\displaystyle x$ -axis. $\displaystyle x^3$ rises everywhere, so $\displaystyle -2x^3$ falls everywhere.
$\displaystyle {p}={1}$ , $\displaystyle {k}={4}$ : the point $\displaystyle (0,0)$ moves to $\displaystyle (1,4)$ .

\displaystyle y=x^{3}

\displaystyle y=-2(x-1)^{3}+4

\displaystyle C=(1,4)

Exercise 1

The board below shows the graph of $\displaystyle {f(x)}={x^3-3x}$ , whose extremum points are $\displaystyle (-1,2)$ and $\displaystyle (1,-2)$ .
The graph of $\displaystyle g$ was made from the graph of $\displaystyle f$ by a shift.
Write $\displaystyle g(x)$ in a row of the board.

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

\displaystyle g(x)=\,?

Exercise 2

The function is $\displaystyle {f(x)}={x^2-2x-3}$ , and its zeros are $\displaystyle {x}={-1}$ and $\displaystyle {x}={3}$ .

Write $\displaystyle f(2x)$ without brackets.

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

/ – 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

Find the zeros of $\displaystyle f(2x)$ .

How do the zeros of $\displaystyle f(2x)$ come from the zeros of $\displaystyle f$ ?
Explain by the squeeze.

Exercise 3

The function is $\displaystyle {f(x)}={x^3-x^2}$ .

Work out $\displaystyle f(-x)$ .

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

/ – 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

The board shows the graphs of $\displaystyle f$ , of $\displaystyle -f(x)$ and of $\displaystyle f(-x)$ .
Write in rows of the board the two functions whose graphs are marked “?”.

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

\displaystyle g(x)=\,?

\displaystyle h(x)=\,?

Exercise 4

The graph on the board was made from the graph of $\displaystyle {y}={x^4}$ by a reflection and a shift.
Write its function in a row of the board.

\displaystyle f(x)=\,?

Exercise 5

The graph on the board was made from the graph of $\displaystyle {y}={|x|}$ by a stretch and shifts.

Write its function in a row of the board.

\displaystyle f(x)=\,?

Work out its zeros.

Exercise 6

The graph of $\displaystyle f$ has one maximum point, $\displaystyle (2,5)$ , and one zero, $\displaystyle {x}={-1}$ .
Write the extremum point and the zero of each function, and say what kind of extremum point it is:

$\displaystyle {g(x)}={f(x+4)}$
$\displaystyle {h(x)}={-f(x)}$
$\displaystyle {k(x)}={f(2x)}$
$\displaystyle {m(x)}={3f(x)}$

Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
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 properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

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

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
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 instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

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
$\displaystyle y-y_1=m(x-x_1)$

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
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$