# How do you graph the function, label the vertex, axis of symmetry, and x-intercepts. x-4 = 1/4 (y+1)^2?

Jul 11, 2015

The vertex is at ($4 , - 1$).
The axis of symmetry is $y = - 1$.
The $x$-intercept is ($\frac{17}{4} , 0$).
There are no $y$-intercepts.

#### Explanation:

The standard form for the equation of a parabola is

$y = {a}^{2} + b x + c$

x−4 = 1/4(y+1)^2

We must get this into standard form.

x−4=1/4(y^2 + 2y + 1)

x−4=1/4y^2 + 1/4(2y)+ 1/4

x−4=1/4y^2 + 1/2y + 1/4

$x = \frac{1}{4} {y}^{2} + \frac{1}{2} y + \frac{1}{4} + 4$

$x = \frac{1}{4} {y}^{2} + \frac{1}{2} y + \frac{17}{4}$

This is standard form, but with $x$ and $y$ interchanged.

We are going to get a sideways parabola.

$a = \frac{1}{4}$, $b = \frac{1}{2}$, and $c = \frac{17}{4}$.

Vertex

Since $a > 0$, the parabola opens to the right.

The $y$-coordinate of the vertex is at

y = –b/(2a) = -(1/2)/(2×(1/4)) = -(1/2)/(1/2) = -1.

Insert this value of $x$ back into the equation.

$x = \frac{1}{4} {y}^{2} + \frac{1}{2} y + \frac{17}{4}$

x=1/4(-1)^2 + 1/2(-1) + 17/4 = 1/4×1 -1/2 +17/4

$x = \frac{1}{4} - \frac{1}{2} + \frac{17}{4} = \frac{1 - 2 + 17}{4} = \frac{16}{4} = 4$

The vertex is at ($4 , - 1$).

Axis of symmetry

The axis of symmetry must pass through the vertex, so

The axis of symmetry is $y = - 1$.

$x$-intercept

To find the $x$-intercept, we set $y = 0$ and solve for $x$.

x=1/4y^2 + 1/2y + 17/4 = 1/4×0^2 + 1/2×0 + 17/4 = 0 + 0 + 17/4 = 17/4

The $x$-intercept is at ($\frac{17}{4} , 0$).

$y$-intercepts

To find the $y$-intercepts, we set $x = 0$ and solve for $y$.

$x = \frac{1}{4} {y}^{2} + \frac{1}{2} y + \frac{17}{4}$

$0 = \frac{1}{4} {y}^{2} + \frac{1}{2} y + \frac{17}{4}$

$0 = {y}^{2} + 2 y + 17$

y = (-b ± sqrt(b^2-4ac))/(2a)

The discriminant D = b^2 – 4ac = 4^2 – 4×1×17= 8 – 68 = -60

Since $D < 0$, there are no real roots.

There are no $y$-intercepts.

Graph

Now we prepare a table of $x$ and $y$ values.

The axis of symmetry passes through $y = - 1$.

Let's prepare a table with points that are 5 units on either side of the axis, that is, from $y = - 6$ to $y = 4$.

Plot these points.

And we have our graph. The red line is the axis of symmetry.