Translation of a Conic Section

Key Questions

How do you translate a graph to the left or right?

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 $(0, 0)$ $(0, 0 )$

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

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

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

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

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

$y = x$ $y = x$

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

$y = x - 5$ $y = x - 5$

seph · 2 · Oct 25 2014
What does it mean to translate a graph?

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

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

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

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

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

Truong-Son N. · 1 · Jun 12 2015
How do you translate a graph up or down?

You can translate any function, $y = f (x)$ $y=f(x)$ using:

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

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

Here is an example:

$f (x) = x^{2} + 2 x$ $f(x)=x^2+2x$

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

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

If we have a different example:

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

It still works!

Amory W. · · Sep 22 2014

Questions

Translation and Rotation of Axis

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Translation of a Conic Section

Key Questions

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Translation and Rotation of Axis