Shifting and reflecting a graph

Algebra and calculus, grade 10 (4 units) Lesson 6 of 20Learning

Shifting and reflecting a graph

A vertical shift

Adding a number to a function adds it to every value, and the whole graph moves up or down:

\displaystyle y=f(x)+k

$\displaystyle k>0$ – the graph moves $\displaystyle k$ units up.
$\displaystyle k<0$ – the graph moves $\displaystyle |k|$ units down.

On the board: $\displaystyle {f(x)}={x^2}$ , and two shifts of it (the dashed ones).

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

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

\displaystyle h(x)=x^{2}-2

$\displaystyle {g(x)}={x^2+3}$ – the same parabola, 3 units up: the vertex has moved from $\displaystyle (0,0)$ to $\displaystyle (0,3)$ .
$\displaystyle {h(x)}={x^2-2}$ – 2 units down: the vertex at $\displaystyle (0,-2)$ .
The shape of the graph does not change, and neither does the domain.

A horizontal shift

Putting $\displaystyle x-p$ in place of $\displaystyle x$ moves the graph to the right or to the left:

\displaystyle y=f(x-p)

$\displaystyle p>0$ – the graph moves $\displaystyle p$ units to the right.
$\displaystyle p<0$ – the graph moves $\displaystyle |p|$ units to the left: $\displaystyle f(x+3)$ is a shift of 3 units to the left.

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

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

\displaystyle g(x)=\sqrt{x-2}

\displaystyle (4,2)

\displaystyle (6,2)

The shift goes against the sign, and this is why:

$\displaystyle g$ takes at $\displaystyle {x}={6}$ the value $\displaystyle f$ takes at $\displaystyle {x}={4}$ : $\displaystyle {g(6)}={\sqrt{6-2}}={\sqrt{4}}={2}={f(4)}$ .
So every value of $\displaystyle f$ reaches $\displaystyle g$ two units later – to the right.
The domain moves too: $\displaystyle \sqrt{x}$ is defined for $\displaystyle x\geq0$ , and $\displaystyle \sqrt{x-2}$ for $\displaystyle x\geq2$ .

Reflections in the axes

A minus before the function turns every value into its opposite; a minus before $\displaystyle x$ puts the opposite of every $\displaystyle x$ in its place:

\displaystyle y=-f(x)

\displaystyle y=f(-x)

On the board: $\displaystyle {f(x)}={\sqrt{x}}$ , and its two reflections (the dashed ones).

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

\displaystyle g(x)=-\sqrt{x}

\displaystyle h(x)=\sqrt{-x}

$\displaystyle {g(x)}={-\sqrt{x}}$ – the reflection in the $\displaystyle x$ -axis: what was above the axis is below it.
$\displaystyle {h(x)}={\sqrt{-x}}$ – the reflection in the $\displaystyle y$ -axis: what was to the right of the axis is to its left. $\displaystyle \sqrt{-x}$ is defined for $\displaystyle x\leq0$ .

Where a point moves

Every transformation moves every point of the graph in the same way. A point $\displaystyle (a,b)$ on the graph of $\displaystyle {y}={f(x)}$ moves:

The new function	The new point	What happened
$\displaystyle {y}={f(x)+k}$	$\displaystyle (a,\ b+k)$	only $\displaystyle y$ changes
$\displaystyle {y}={f(x-p)}$	$\displaystyle (a+p,\ b)$	only $\displaystyle x$ changes
$\displaystyle {y}={-f(x)}$	$\displaystyle (a,\ -b)$	$\displaystyle y$ changes sign
$\displaystyle {y}={f(-x)}$	$\displaystyle (-a,\ b)$	$\displaystyle x$ changes sign

A point on the $\displaystyle x$ -axis stays where it is in the reflection in the $\displaystyle x$ -axis.
A point on the $\displaystyle y$ -axis stays where it is in the reflection in the $\displaystyle y$ -axis.

Two shifts together

$\displaystyle {g(x)}={\sqrt{x+1}-2}$ .
How do you get the graph of $\displaystyle g$ from the graph of $\displaystyle {f(x)}={\sqrt{x}}$ ?

The example below reads the two shifts from the formula, and finds where the graph starts.

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

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

\displaystyle (-1,-2)

$\displaystyle \sqrt{x+1}$ – a shift of one unit to the left (here $\displaystyle {p}={-1}$ ).
$\displaystyle -2$ – a shift of 2 units down.
The graph of $\displaystyle \sqrt{x}$ starts at $\displaystyle (0,0)$ ; after the two shifts it starts at $\displaystyle {(0-1,\ 0-2)}={(-1,-2)}$ .
The domain of $\displaystyle g$ : $\displaystyle x+1\geq0$ , that is $\displaystyle x\geq-1$ .

Exercise 1

On the board: $\displaystyle {f(x)}={\sqrt{x}}$ , and three graphs whose formulas are hidden.
Each of them is a shift of $\displaystyle f$ : the formulas are $\displaystyle \sqrt{x-1}$ , $\displaystyle \sqrt{x+1}$ and $\displaystyle \sqrt{x}-1$ .
Write the formula of each graph in a row of its own.

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

\displaystyle g(x)=\,?

\displaystyle h(x)=\,?

\displaystyle k(x)=\,?

Exercise 2

$\displaystyle {f(x)}={x^3}$ .
The graph of $\displaystyle g$ is the graph of $\displaystyle f$ , shifted 2 units to the right and one unit up.

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

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

The point $\displaystyle (1,1)$ is on the graph of $\displaystyle f$ .
To which point on the graph of $\displaystyle g$ does it move?

Exercise 3

On the board – the function $\displaystyle f$ ; its graph is made of straight pieces.
The points $\displaystyle A(-2,-1)$ and $\displaystyle B(1,1)$ are on the graph.

\displaystyle f(x)

\displaystyle A=(-2,-1)

\displaystyle B=(1,1)

$\displaystyle {g(x)}={f(x)-4}$ .
To which points on the graph of $\displaystyle g$ do $\displaystyle A$ and $\displaystyle B$ move?

$\displaystyle {h(x)}={f(x+3)}$ .
To which points on the graph of $\displaystyle h$ do $\displaystyle A$ and $\displaystyle B$ move?

$\displaystyle {k(x)}={-f(x)}$ .
To which points on the graph of $\displaystyle k$ do $\displaystyle A$ and $\displaystyle B$ move?
Which point of the graph of $\displaystyle f$ stays where it is?

Exercise 4

$\displaystyle {f(x)}={x^2-4x}$ .
On the board: $\displaystyle f$ , its reflection in the $\displaystyle x$ -axis and its reflection in the $\displaystyle y$ -axis.
Write the formula of each reflection in a row of its own.

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

\displaystyle g(x)=\,?

\displaystyle h(x)=\,?

Exercise 5

The graph of $\displaystyle {f(x)}={x^2}$ is moved to the right and down, until its vertex reaches $\displaystyle V(3,-2)$ .
Write the formula of the new parabola on the board.

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

\displaystyle V=(3,-2)

Where does the new parabola decrease, and where does it increase?

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
$\displaystyle f'(x)=0$