Question 204354
In general, if you have a function (let's call it y = f(x)), then the graph of
c(y - k) = f(b(x - h)) will be<ul><li>Shifted/translated right by "h" units</li><li>Shifted/translated up by "k" units</li><li>Compressed horizontally by a factor of "b"</li><li>Be reflected in the y-axis if b < 0</li><li>Be compressed vertically by a factor of "c"</li><li>Be reflected in the x-axis if c < 0</li></ul>
Since you want<ul><li>a translation <i>left</i> of 2, your "h" will be -2.</li><li>a translation <i>down</i> of 5, your "k" will be -5.</li><li>a horizontal <i>expansion</i> factor of 3 (but no horizontal reflexion), your "b" will be 1/3.</li><li>no vertical compression/expansion but you do want a vertical reflexion, you "c" will be -1.</li></ul>
Applying the general form to your function:
{{{c(y - k) = sqrt(b(x - h))}}}
Substituting your values for the parameters:
{{{-1(y - (-5)) = sqrt((1/3)(x - (-2)))}}}
Now all that is left is to simplify and solve for y:
{{{-1(y + 5) = sqrt((1/3)(x + 2))}}}
Rationalize the denominator in the square root:
{{{-1(y + 5) = sqrt((3/9)(x + 2))}}}
{{{-1(y + 5) = sqrt((3)(x + 2))/sqrt(9)}}}
{{{-1(y + 5) = sqrt(3x + 6)/3}}}
Multiply both sides by -1:
{{{y + 5 = - sqrt(3x + 6)/3}}}
Subtract 5 from both sides:
{{{y = -5 - sqrt(3x + 6)/3}}}