We want to find out a few pieces of information about the given function.

What family does the function belong to?
What is the domain of this function?
What range does the function have?

Let's answer these questions!

Function Family

Look carefully at the function rule.

f (x) = - 2 x^{2} + 6

We can see that given function is quadratic. Therefore, it belongs to the family of quadratic functions, with a parent function of $y = x^{2} .$

Domain

Unless specific restrictions are given, the domain of quadratic functions is usually all real numbers.

Domain : - \infty < x < \infty

Range

The range of quadratic functions depends on the

y -

coordinate of the vertex, so let's find this value first.

f (x) = - 2 x^{2} + 6 \Leftrightarrow f (x) = - 2 x^{2} + 0 x + 6

We can see above that

a = - 2,

b = 0,

and

c = 6 .

To calculate the vertex, we need to think of

y

as a function of

x,

as

y = f (x) .

We can write the expression for the vertex by stating the

x -

and

y -

coordinates in terms of

a

and

b .

Vertex : (- \frac{b}{2 a}, f (- \frac{b}{2 a}))

By substituting

- \frac{0}{2 ( - 2 )} = 0

for

x

in the function rule, we can solve for the

y -

coordinate.

f (x) = - 2 x^{2} + 6

Substitute

$x = 0$

f (0) = - 2 (0)^{2} + 6

CalcPow

Calculate power

f (0) = - 2 (0) + 6

ZeroPropMult

Zero Property of Multiplication

f (0) = 6

Since the

y -

coordinate of vertex equals

6,

and the parabola opens downwards (because

a < 0

), the range of this function is all real numbers less than or equal to

6 .

Range : y \leq 6

Checking the Answer

To verify your answer, enter the function rule in the calculator by pushing $Y =$ and writing the function rule in the first row. Next, by pushing $GRAPH,$ the calculator will draw the graph of the function.

Looking at the graph, we can confirm that our solution is correct.

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