Parameter Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Parameter.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A parameter is a quantity in a mathematical expression that remains constant for a particular case but can change between cases. For example, in y = mx + b, the slope m and intercept b are parameters — each choice of m and b gives a different line.

In $y = mx + b$ , $m$ and $b$ are parameters — different values give different lines.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A parameter is held fixed within one case but turned between cases to select different functions.

Common stuck point: The procedure for parameter is the easy part; the trap is treating the parameter as the input variable. Asking "Is this quantity held constant within one case but changed to produce a different member of a family?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this quantity held constant within one case but changed to produce a different member of a family?

Worked Examples

Example 1

medium

In the family of lines

y = mx + 3

, what role does

m

play?

Answer

m

is a parameter that determines the slope of each line in the family.

First step

The equation

y = mx + 3

represents infinitely many lines, all passing through

(0, 3)

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Example 2

hard

For what value of the parameter

k

does

kx + 2 = 6

have the solution

x = 2

Example 3

medium

In the family of lines

y = mx + 3

, find the slope when the line passes through

(2, 7)

Example 4

medium

For what value of

a

does the line

y = ax + 2

pass through the point

(3, 11)

Example 5

medium

y = kx

, find

k

if the line passes through

(4, 10)

Example 6

medium

The line

y = kx + 2

passes through the point

(-3, 8)

. Find

k

Example 7

medium

Find

a

so that the parabola

y = ax^2 - 4x + 1

passes through

(2, 5)

Example 8

medium

In the family

y = a(x - 2)^2 + 3

, find

a

if the parabola passes through

(4, 11)

Example 9

hard

For what values of

k

does the equation

x^2 + kx + 4 = 0

have two distinct real roots?

Example 10

hard

Find

a

and

b

if the parabola

y = ax^2 + bx

passes through

(1, 5)

and

(2, 14)

Example 11

hard

The cost of producing

x

items is modeled by

C(x) = a x + b

, where

a

is the variable cost per item and

b

is the fixed cost. If

C(100) = 700

and

C(200) = 1200

, find

a

and

b

Example 12

challenge

Find all values of the parameter

m

so that the line

y = mx - 1

is tangent to the parabola

y = x^2

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

y = 2x + b

, is

b

a variable or a parameter?

Example 2

medium

The quadratic

f(x) = x^2 - px + 4

has a zero at

x = 1

. Find

p

Example 3

easy

y=mx+b

, name the parameters.

Example 4

easy

y=mx+b

, which symbols are the variables?

Example 5

easy

For the line

y=2x+1

, what is the slope parameter?

Example 6

easy

f(x)=ax^2

, if

a

changes from 1 to 3, does the curve change?

Example 7

easy

y=mx+b

with

m=0

, what kind of line results?

Example 8

easy

In the circle

x^2+y^2=r^2

, which symbol is the parameter?

Example 9

easy

Can you 'solve for' the parameter

m

y=mx+b

like an unknown?

Example 10

easy

y=mx+b

, for one fixed line, does

b

stay constant as

x

changes?

Example 11

medium

The line

y=mx+2

passes through

(3,8)

. Find the parameter

m

Example 12

medium

For

y=ax^2

passing through

(2,12)

, find

a

Example 13

medium

Two points

(0,1)

and

(2,5)

lie on

y=mx+b

. Find both parameters.

Example 14

medium

y=mx+b

, describe how the graph moves as

b

increases with

m

fixed.

Example 15

medium

A spring model

F=kx

has parameter

k

. If

F=10

when

x=2

, find

k

Example 16

medium

For the family

y=x^2+c

, which member passes through the origin?

Example 17

medium

y=mx+b

, a problem fixes

m=3

throughout. Is changing

m

to 4 partway valid?

Example 18

challenge

The parabola

y=ax^2+bx+c

has vertex at

(1,-2)

and passes through

(0,0)

. With

a=2

, find

b

and

c

Example 19

challenge

For which parameter value

k

does

kx+2y=6

pass through

(2,1)

, and what stays variable?

Example 20

challenge

The family

y=a(x-2)^2

all share a common point regardless of

a

. Find it.

Example 21

medium

The line

y=mx-1

passes through

(2,5)

. Find

m

Example 22

medium

y=a(x-1)^2

, find

a

if the curve passes through

(3,8)

Example 23

easy

In the family

y = ax^2

, what does changing

a

do to the graph?

Example 24

easy

y = mx + 7

, what does the parameter

m

control?

Example 25

easy

In the equation

y = 3x + b

, find

b

if the line passes through

(0, -4)

Example 26

medium

In the family of parabolas

y = (x - h)^2

, what does the parameter

h

control?

Example 27

medium

For what value of

c

does the equation

x^2 - 6x + c = 0

have a single (repeated) root?

Example 28

medium

In the family

y = mx + b

, fix

b = 3

and vary

m

. What do all such lines have in common?

Example 29

medium

Solve for the parameter

b

y = -2x + b

passes through

(4, 1)

Example 30

hard

In the line

y = mx - 1

, for what slope parameter

m

is the line perpendicular to

y = 4x + 3

Example 31

hard

For what value of

k

does the system

\begin{cases} y = 2x + 1 \\ y = kx + 5 \end{cases}

have no solution?

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Related Concepts

Variables Linear Functions Parametric Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

variableslinear functions