# Translation of a Conic Section

## Key Questions

• When translating a graph to the left or to the right, it means moving the entire graph to the left or to the right of its current location, which is usually relative to $\left(0 , 0\right)$

For example, let's say you have a circle centered at $\left(0 , 0\right)$ with radius 2. Its standard equation would be

${x}^{2} + {y}^{2} = 4$

Now, let's say we translate the circle 5 units to the left. Your circle will now be centered at (-5, 0) with the radius still equal to 2. Its new standard equation would be

${\left(x + 5\right)}^{2} + {y}^{2} = 0$

This is also true for linear equations.
For example, let's say you have a line with slope 1 with the x-intercept equal to 0. Its equation will be

$y = x$

When we translate the line 3 units to the right, its slope will remain the same, but its x-intercept will now be 3. Its new equation will be

$y = x - 5$

• Translations are simply horizontal and/or vertical shifts. So, $y = \sqrt{x} - 2$ is shifted 2 units down from $y = \sqrt{x}$, while $y = \sqrt{x - 2}$ is shifted 2 units right of $y = \sqrt{x}$.

$y = \sqrt{x}$:
graph{sqrtx [-10, 10, -5, 5]}

$y = \sqrt{x} - 2$:
graph{sqrtx - 2 [-10, 10, -5, 5]}

$y = \sqrt{x - 2}$:
graph{sqrt(x-2) [-10, 10, -5, 5]}

$y = \sqrt{x - 2} - 2$:
graph{sqrt(x-2) - 2 [-10, 10, -5, 5]}

• You can translate any function, $y = f \left(x\right)$ using:

$y = f \left(x - h\right) + k$ or
$y - k = f \left(x - h\right)$

If $h$ is positive, the graph will translate to the right.
If $h$ is negative, the graph will translate to the left.
If $k$ is positive, the graph will translate up.
If $k$ is negative, the graph will translate down.

Here is an example:

$f \left(x\right) = {x}^{2} + 2 x$

If we want to translate this up 4 units, then we have:

$y = f \left(x\right) + 4$
$y = {x}^{2} + 2 x + 4$

If we have a different example:

$f \left(x\right) = \frac{1}{x}$
$y = f \left(x\right) + 4$
$y = \frac{1}{x} + 4$

It still works!