Function Families Math Example 1

Follow the full solution, then compare it with the other examples linked below.

easy

The family of quadratics

f(x)=ax^2

(with

a\neq0

) all share the same vertex at the origin. Describe how changing

a

from

1

4

-2

affects the graph.

1
$a=1$ : standard parabola, opens up, vertex at origin. $f(3)=9$ .
2
$a=4$ : opens up, narrower (vertically stretched by $4$ ). $f(3)=36$ .
3
$a=-2$ : opens down (reflected), vertically stretched by $2$ . $f(3)=-18$ . All share vertex $(0,0)$ and $x$ -intercept at $x=0$ . Parameter $a$ controls direction and width.

Answer

a>0

: opens up;

a<0

: opens down;

|a|>1

: narrower;

|a|<1

: wider

A function family is a set of functions sharing a common form, differing only in parameter values. Varying

a

ax^2

produces every possible parabola with vertex at the origin, illustrating how one parameter controls an entire continuum of shapes.

About Function Families

A function family is a group of functions sharing the same general form and behavior, differing only in the values of one or more parameters.

The family [formula] has a vertical asymptote at [formula] for each parameter [formula]. Analyze the

In the family [formula] (exponential), how does the graph change as [formula] varies? Compare [formu

The family [formula] for integer [formula] has different behavior depending on whether [formula] is