A function's formula

A function’s formula

A function’s rule is written short as a formula: the rule “multiply by 3 and subtract 5” is written $\displaystyle {y}={3x-5}$ .

On one side $\displaystyle y$ , the dependent variable.
On the other side an algebraic expression with $\displaystyle x$ , the independent variable.
Sometimes the function is given a name, say $\displaystyle f$ , and written $\displaystyle {f(x)}={3x-5}$ . It is read “ $\displaystyle f$ of $\displaystyle x$ ”.

Substituting in the formula

A negative number goes in with brackets:

\displaystyle 3\cdot(-2)

, not

\displaystyle 3-2

. In a power too: if

\displaystyle {y}={x^{2}+1}

and

\displaystyle {x}={-3}

, then

\displaystyle {y}={(-3)^{2}+1}={10}

To find the $\displaystyle y$ of a given $\displaystyle x$ , substitute it in the formula: write the number in place of $\displaystyle x$ , and work it out.
In the function $\displaystyle {y}={3x-5}$ :

$\displaystyle {x}={4}$ : $\displaystyle {y}={3\cdot4-5}={12-5}={7}$ .
$\displaystyle {x}={-2}$ : $\displaystyle {y}={3\cdot(-2)-5}={-6-5}={-11}$ .
$\displaystyle {x}={0}$ : $\displaystyle {y}={3\cdot0-5}={-5}$ .

Finding

\displaystyle x

when

\displaystyle y

is known

When $\displaystyle y$ is known, substituting it in the formula gives an equation in $\displaystyle x$ .
At which $\displaystyle x$ does the function $\displaystyle {y}={3x-5}$ take the value 10?

Substitute $\displaystyle {y}={10}$ : $\displaystyle {3x-5}={10}$ .
Add 5 to both sides: $\displaystyle {3x}={15}$ .
Divide by 3: $\displaystyle {x}={5}$ . So the point $\displaystyle (5,10)$ is on the graph.

From a formula to a table and a graph

To draw a function’s graph from its formula:

Choose some values of $\displaystyle x$ .
Substitute each of them in the formula, and write a table.
Mark the points in the coordinate system, and join them with a line.

The function $\displaystyle {y}={x^{2}-3}$ :

$\displaystyle x$	$\displaystyle -3$	$\displaystyle -2$	$\displaystyle -1$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 3$
$\displaystyle y$	$\displaystyle 6$	$\displaystyle 1$	$\displaystyle -2$	$\displaystyle -3$	$\displaystyle -2$	$\displaystyle 1$	$\displaystyle 6$

\displaystyle y=x^{2}-3

\displaystyle (-3,6)

\displaystyle (-2,1)

\displaystyle (-1,-2)

\displaystyle (0,-3)

\displaystyle (1,-2)

\displaystyle (2,1)

\displaystyle (3,6)

This graph is not a straight line, so the points are joined by a curved line. To know its shape between two points, substitute more values of $\displaystyle x$ between them: for example $\displaystyle {x}={0.5}$ gives $\displaystyle {y}={0.25-3}={-2.75}$ .

Exercise 1

The function’s formula: $\displaystyle {y}={4x-7}$ .
Find $\displaystyle y$ for each of the values of $\displaystyle x$ .

$\displaystyle {x}={3}$

/ – 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

$\displaystyle {x}={-2}$

/ – 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

$\displaystyle {x}={0}$

/ – 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

$\displaystyle {x}={\frac{1}{2}}$

/ – 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 2

The function’s formula: $\displaystyle {y}={5x+2}$ .
Find the $\displaystyle x$ at which the function takes each of the values.
Write an equation and solve it.

$\displaystyle {y}={27}$

$\displaystyle {y}={-8}$

Exercise 3

The function’s formula: $\displaystyle {y}={x^{2}-2x}$ .

Make a table of values for $\displaystyle {x}={-1, 0, 1, 2, 3}$ .
Mark the points of the table in the coordinate system.
Show the function’s graph, and check: are all your points on it?

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

Exercise 4

The phone plan costs 30 shekels a month.
Each minute of a call abroad costs 0.5 shekels more.

Write a formula: $\displaystyle y$ – the bill in shekels in a month in which you talked abroad for $\displaystyle x$ minutes.
How much do you pay for 25 minutes?
One month the bill was 62 shekels. How many minutes did you talk abroad? Write an equation and solve it.

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Formulas: algebra, grade 8

Open as a page

The linear function

A linear function – $\displaystyle m$ the slope; the line crosses the $\displaystyle y$ -axis at $\displaystyle (0,b)$
$\displaystyle y=mx+b$
The slope of the line through $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$
Parallel lines have the same slope
$\displaystyle m_1=m_2$
A line through the origin – $\displaystyle {b}={0}$
$\displaystyle y=mx$
A horizontal line – the slope is 0
$\displaystyle y=b$

Word problems

$\displaystyle p$ percent of $\displaystyle x$
$\displaystyle \frac{p}{100}\cdot x$
$\displaystyle s$ the distance, $\displaystyle v$ the speed, $\displaystyle t$ the time
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$

Inequalities

Adding or subtracting the same number on both sides – the sign stays
$\displaystyle a+c<b+c$ $\displaystyle a<b$
Multiplying or dividing by a positive number – the sign stays
$\displaystyle a\cdot c<b\cdot c$ $\displaystyle a<b,\ c>0$
Multiplying or dividing by a negative number – the sign turns round
$\displaystyle a\cdot c>b\cdot c$ $\displaystyle a<b,\ c<0$