Equations, the graph, shifts and reflection

Algebra, grade 9 Lesson 18 of 26Learning

Equations, the graph, shifts and reflection

The graph of a function shows the solutions of its equations, and a change in the function moves its graph:

$\displaystyle {f(x)}={k}$ – where the graph meets the horizontal line $\displaystyle {y}={k}$ ;
$\displaystyle f(x)+k$ and $\displaystyle f(x-b)$ – the graph shifted up or down, right or left;
$\displaystyle -f(x)$ – the graph reflected in the $\displaystyle x$ -axis.

The equation

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

A solution is a value of

\displaystyle x

\displaystyle {x}={2}

, not the point

\displaystyle (2,4)

The solutions of $\displaystyle {f(x)}={4}$ are the $\displaystyle x$ values of the points where the graph of $\displaystyle f$ meets the line $\displaystyle {y}={4}$ (dashed). Here there are two, $\displaystyle A$ and $\displaystyle B$ , so the equation has two solutions: $\displaystyle {x}={-4}$ and $\displaystyle {x}={2}$ .

\displaystyle f(x)=-\frac{1}{4}x^{3}+3x

\displaystyle y=4

\displaystyle A=(-4,4)

\displaystyle B=(2,4)

The number of solutions

The number of solutions of $\displaystyle {f(x)}={k}$ is the number of meeting points of the graph and the line $\displaystyle {y}={k}$ . Moving the line up or down on the graph above:

$\displaystyle {f(x)}={6}$ – one solution;
$\displaystyle {f(x)}={0}$ – three solutions (one of them is $\displaystyle {x}={0}$ );
$\displaystyle {f(x)}={4}$ – two solutions: at $\displaystyle B$ the line only touches the graph.

Exercise 1

Answer from the graph of $\displaystyle f$ above.

Solve the equation $\displaystyle {f(x)}={-4}$ .
How many solutions has the equation $\displaystyle {f(x)}={2}$ ? And $\displaystyle {f(x)}={-6}$ ?
For which values of $\displaystyle k$ has the equation $\displaystyle {f(x)}={k}$ exactly two solutions?

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

/ – 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 {f(x)}={-6}

/ – 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

Vertical shift:

\displaystyle f(x)+k

The dashed graph, $\displaystyle h(x)+2$ , is the graph of the function $\displaystyle h$ (solid) moved 2 up: every point $\displaystyle (x,y)$ becomes $\displaystyle (x,y+2)$ . The minimum points $\displaystyle (-2,-4)$ and $\displaystyle (2,-4)$ become $\displaystyle (-2,-2)$ and $\displaystyle (2,-2)$ , and the maximum point $\displaystyle (0,0)$ becomes $\displaystyle (0,2)$ .

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

\displaystyle y=\frac{1}{4}x^{4}-2x^{2}+2

In general, $\displaystyle f(x)+k$ moves the graph $\displaystyle k$ up when $\displaystyle k>0$ , and down when $\displaystyle k<0$ : $\displaystyle f(x)-3$ is 3 down.

Horizontal shift:

\displaystyle f(x-b)

Careful:

\displaystyle f(x+3)

moves the graph 3 to the left, since

\displaystyle {x+3}={x-(-3)}

The graph of $\displaystyle {g(x)}={h(x-2)}$ (dashed) is the graph of $\displaystyle h$ moved 2 to the right. The reason: $\displaystyle {g(4)}={h(2)}$ – $\displaystyle g$ takes at $\displaystyle {x}={4}$ the value $\displaystyle h$ takes at $\displaystyle {x}={2}$ , two places to the left. Every point $\displaystyle (x,y)$ becomes $\displaystyle (x+2,y)$ : the maximum point $\displaystyle (0,0)$ becomes $\displaystyle (2,0)$ .

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

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

$\displaystyle f(x-b)$ with $\displaystyle b>0$ : $\displaystyle b$ to the right.
$\displaystyle f(x+b)$ with $\displaystyle b>0$ : $\displaystyle b$ to the left.

Reflection in the

\displaystyle x

-axis:

\displaystyle -f(x)

The graph of $\displaystyle -h(x)$ (dashed) is the graph of $\displaystyle h$ reflected in the $\displaystyle x$ -axis – flipped over it, as in a mirror: every point $\displaystyle (x,y)$ becomes $\displaystyle (x,-y)$ .

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

\displaystyle y=-\frac{1}{4}x^{4}+2x^{2}

A minimum point becomes a maximum point: $\displaystyle (-2,-4)$ becomes $\displaystyle (-2,4)$ .
Increasing becomes decreasing, and positive becomes negative.
The intersection points with the $\displaystyle x$ -axis stay where they are.

The properties in a shift: kept and changed

A shift moves the whole graph without changing its shape, so the number and the kind of the extremum points stay. The rest:

Property	$\displaystyle f(x)+k$	$\displaystyle f(x-b)$
extremum points	$\displaystyle y$ changes by $\displaystyle k$	$\displaystyle x$ changes by $\displaystyle b$
maximum and minimum values	change by $\displaystyle k$	the same
increasing and decreasing domains	the same	move by $\displaystyle b$
intersection points with the axes	change	change

Exercise 2

The function $\displaystyle f$ has a minimum point $\displaystyle (-1,-3)$ and a maximum point $\displaystyle (2,5)$ .
It is increasing for $\displaystyle -1<x<2$ , and decreasing for $\displaystyle x<-1$ and for $\displaystyle x>2$ .
Write the extremum points and the increasing and decreasing domains of:

$\displaystyle f(x)+4$

$\displaystyle f(x-3)$

$\displaystyle f(x+1)$

$\displaystyle -f(x)$

Exercise 3

The solid graph is the function $\displaystyle f$ from the start of the page, with its extremum points $\displaystyle A$ and $\displaystyle B$ .
The dashed graph is a function $\displaystyle g$ : the graph of $\displaystyle f$ shifted.

Write $\displaystyle g(x)$ with $\displaystyle f$ .
How many solutions has the equation $\displaystyle {g(x)}={2}$ ?

\displaystyle f(x)=-\frac{1}{4}x^{3}+3x

\displaystyle g(x)=\,?

\displaystyle A=(-2,-4)

\displaystyle B=(2,4)

\displaystyle C=(-3,-6)

\displaystyle D=(1,2)

\displaystyle {g(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

Exercise 4

Sketch the graph of $\displaystyle h(x)-4$ for the function $\displaystyle h$ above (its minimum points $\displaystyle (-2,-4)$ and $\displaystyle (2,-4)$ , its maximum point $\displaystyle (0,0)$ ).
Then, on another grid, sketch $\displaystyle h(x+3)$ .

Formulas: algebra, grade 9

Open as a page

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