Exercise 26 Page 410

To algebraically determine the inverse of the given relation, we exchange

x

and

y

and solve for

y .

y = 4 x^{2} - 2 ⟶ switch x = 4 y^{2} - 2

The result of isolating

y

in the new equation will be the inverse of the given function.

x = 4 y^{2} - 2

▼

Solve for

y

AddEqn

$LHS + 2 = RHS + 2$

x + 2 = 4 y^{2}

DivEqn

$LHS / 4 = RHS / 4$

\frac{x + 2}{4} = y^{2}

SqrtEqn

$LHS = RHS$

\pm \frac{x + 2}{4} = y

RearrangeEqn

Rearrange equation

y = \pm \frac{x + 2}{4}

Now we have the inverse of the given function.

y = \pm \frac{x + 2}{4}

Graphing the Relation

Because the given function is a parabola, to graph it we should first determine its vertex. Notice that the function is in standard form, so let's start with highlighting the coefficients.

\underline{Standard Form} y = a x^{2} + b x + c \underline{Function} y = 4 x^{2} + 0 x + (- 2)

In this form, if

a

is positive, the parabola opens upward. If

a

is negative, the parabola opens downward. Since

4 > 0,

the parabola of this function opens upward. To find the vertex, we first need to find the

x

coordinate of the vertex.

x = - \frac{b}{2 a}

Let's find it!

x = - \frac{b}{2 a}

SubstituteII

$a = 4$ , $b = 0$

x = - \frac{0}{2 ( 4 )}

▼

Simplify

Multiply

x = - \frac{0}{8}

CalcQuot

Calculate quotient

x = 0

The

x

coordinate of the vertex is

x = 0 .

By substituting

0

for

x

into the function, we can find its

y

coordinate.

y = 4 x^{2} - 2

Substitute

$x = 0$

y = 4 (0)^{2} - 2

▼

Simplify

CalcPow

Calculate power

y = 4 \cdot 0 - 2

ZeroPropMult

Zero Property of Multiplication

y = 0 - 2

SubTerm

Subtract term

y = - 2

Thus, the vertex of the parabola is

(0, - 2) .

To graph the parabola let's choose two more points, one on either side of the vertex. Let's use

x = - 1

and

x = 1 .

By substituting these coordinates into the function, we can find the

y

coordinates.

$x$	$y = 4 x^{2} - 2$	$y$	Point
$x = - 1$	$y = 4 (- 1)^{2} - 2$	$y = 2$	$(- 1, 2)$
$x = 0$	$y = 4 (0)^{2} - 2$	$y = - 2$	$(0, - 2)$
$x = 1$	$y = 4 (1)^{2} - 2$	$y = 2$	$(1, 2)$

Let's plot the points and connect them to graph the parabola.

Graphing the Inverse of the Relation

Finally, we can graph the inverse of the function by reflecting the parabola across $y = x .$ This means that we should interchange the $x$ and $y$ coordinates of the points that are on the parabola.

Points	Reflection across $y = x$
$(- 1, 2)$	$(2, - 1)$
$(0, - 2)$	$(- 2, 0)$
$(1, 2)$	$(2, 1)$