Exercise 46 Page 348

Before we begin, note that in the given equation the variable that is raised to the second power is x. x^2=3y Therefore, the axis of symmetry of the parabola is a vertical line.

Finding the Desired Information

y=1/4 p(x- h)^2+ k Let's rearrange the given equation.

x^2=3y

MultEqn

LHS * 1/3=RHS* 1/3

1/3x^2=y

RearrangeEqn

Rearrange equation

y=1/3x^2

We need to identify the values of k, h, and p. Let's start with p. To do so, we will solve the equation 14 p= 13. We set it equal to 13 because it is the coefficient of the x^2 term.

1/4p=1/3

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Solve for p

MultEqn

LHS * 3=RHS* 3

1/4p(3)=1

MoveRightFacToNumOne

1/b* a = a/b

3/4p=1

MultEqn

LHS * p=RHS* p

3/4=p

RearrangeEqn

Rearrange equation

p=3/4

Knowing that p= 34, we can rewrite the equation. y=1/3x^2 ⇕ y=1/4( 3/4)(x- 0)^2+ 0 Now we have that k= 0, h= 0, and p= 34. By recalling the corresponding formulas, we can find the focus, directrix, and axis of symmetry of the parabola.

	Focus	Directrix	Axis of Symmetry
Formula	( h, k+ p)	y= k- p	x= h
Value	( 0, 0+ 3/4) ⇓ (0,3/4)	y= 0- 3/4 ⇓ y=- 3/4	x= 0

Drawing the Parabola

Now, let's draw the parabola using the obtained information.