11.4: Shifted Conics

The first three sections describe the algebra behind ellipses, parabolas, and hyperbolas.

It is time to learn how to shift them.

Shifting Graphs of Equations

Suppose $h, k > 0$. For an equation in $x$ and $y$, the following shifts are possible:
Replacement How the graph is shifted
$x$ replaced with $x - h$ Right $h$ units
$x$ replaced with $x + h$ Right $h$ units
$y$ replaced with $y - k$ Up $k$ units
$y$ replaced with $y + k$ Down $k$ units

Shifted Ellipses

To shift an ellipse:

  1. Identify if the parent ellipse is either \[\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \qquad \text{or} \qquad \dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\]
  2. Then, identify the shifts.

Sketch a graph of the ellipse \[\dfrac{(x+1)^2}{4} + \dfrac{(y-2)^2}{9} = 1\] Also include the foci in the sketch.
Sketch a graph of the ellipse \[x^2 + \dfrac{(y+2)^2}{4} = 1\] Also include the foci in the sketch.

Shifted Parabolas

Shifting a parabola is similar to shifting an ellipse:

  1. Identify if the parent parabola is either \[y^2 = 4px \qquad \text{or} \qquad x^2 = 4py\]
  2. Then, identify the shifts.

Sometimes, we will need a method called "Completing the Square."

Completing the Square

Recall that \[(A + B)^2 = A^2 + 2AB + B^2\]

If $B^2$ is missing, you can add and subtract it to be able to factor it. In general:

To complete the square for $x^2 + bx$, add and subtract $\left(\dfrac{b}{2}\right)^2$. Witness: \begin{align} x^2 + bx &= x^2 + bx + \left(\dfrac{b}{2}\right)^2 - \left(\dfrac{b}{2}\right)^2 \\&= \left(x + \dfrac{b}{2}\right)^2 - \dfrac{b^2}{4} \end{align}
Complete the square for $x^2 + 4x$.

Notice that completing the square requires the coefficient of $x^2$ to be one.

Complete the square for $2x^2 + 6x$.

Now we are ready to shift parabolas.

Sketch a graph of the parabola \[x^2 - 4x = 8y - 28\] Include the vertex, focus, and directrix.
Sketch a graph of the parabola \[y^2 - 6y - 12x + 33 = 0\] Include the vertex, focus, and directrix.

Shifted Hyperbolas

To shift a hyperbola:

  1. Identify if the parent hyperbola is either \[\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\qquad \text{or} \qquad \dfrac{y^2}{a^2} - \dfrac{x^2}{b^2} = 1\]
  2. Then, identify the shifts.

Consider the hyperbola \[9x^2 - 72x - 16y^2 - 32y = 16\]
  1. Complete the square to get the shifted form.
  2. Sketch a graph. Include the vertices, foci, and asymptotes.
Sketch a graph of the hyperbola \[\dfrac{(x+1)^2}{9} - \dfrac{(y-3)^2}{16} = 1\] Include the vertices, foci, and asymptotes.