Parameter Examples: 43 Problems with Answers

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.

Read the full concept explanation →

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

1
The equation y=mx+3 represents infinitely many lines, all passing through (0,3).

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

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
In 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=ax2−4x+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 x2+kx+4=0 have two distinct real roots?

Example 10

hard
Find a and b if the parabola y=ax2+bx passes through (1,5) and (2,14).

Example 11

hard
The cost of producing x items is modeled by C(x)=ax+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=x2.

Practice Problems

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

Example 1

easy
In y=2x+b, is b a variable or a parameter?

Example 2

medium
The quadratic f(x)=x2−px+4 has a zero at x=1. Find p.

Example 3

easy
In y=mx+b, name the parameters.

Example 4

easy
In 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
In f(x)=ax2, if a changes from 1 to 3, does the curve change?

Example 7

easy
In y=mx+b with m=0, what kind of line results?

Example 8

easy
In the circle x2+y2=r2, which symbol is the parameter?

Example 9

easy
Can you 'solve for' the parameter m in y=mx+b like an unknown?

Example 10

easy
In 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=ax2 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
In 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=x2+c, which member passes through the origin?

Example 17

medium
In y=mx+b, a problem fixes m=3 throughout. Is changing m to 4 partway valid?

Example 18

challenge
The parabola y=ax2+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
In y=a(x−1)2, find a if the curve passes through (3,8).

Example 23

easy
In the family y=ax2, what does changing a do to the graph?

Example 24

easy
In 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 x2−6x+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 {y=2x+1y=kx+5 have no solution?
← Back to Parameter Formula Explained Practice Mode

Background Knowledge

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

variableslinear functions