If g(x) is the image of y=−2x^2−16x+2 after a translation left 4 units and down 3 units, write the equation of g(x) in the standard form of a quadratic function and describe its graph.

Question

The standard form equation is g(x)= _____

The graph of g(x) is a parabola that opens ________ with a vertex at _______

Answer 1

To do the transformation, we want to have the quadratic in Vertex Form. Completing the Square is a process that will convert this equation into vertex form, A(x - h)² + k where "A" is the Vertical Stretch Factor, "h" is the Horizontal Shift, and "k" is the Vertical Shift. Also, the point (h, k) is the vertex of the parabola.

To Complete the Square:

Step 1: Move the "number" off to the side:

−2x² − 16x + 2

Step 2: Factor Out the number in front of the "x²"

-2(x² + 8x ) + 2

Step 3: Add in a number the "completes the square" and then subtract it out again:

To find out the number you add in, take 1/2 of the number in front of the "x" and then square it. The number in front of the "x" is "+ 8" so half of the is "+ 4" and then square it to get "+ 16" so add in " + 16":

-2(x² + 8x + 16) + 2

But you can't just add 16 to an equation, right? That would be changing it. So we want to subtract that out again to make it equivalent to adding zero. Here's the tricky part though. You didn't just add "+ 16", there is a "- 2" out front that you previously factored out. So, in actuality, you added in "- 32". So to "zero that out" you must add "+ 32" so we put in "+ 32 at the end:

-2(x² + 8x + 16) + 2 + 32

Step 4: Write the part in parenthesis as a square:

To do that, you group with the x the number that is 1/2 of what is in front of the "x" and since "+ 8" is in front of the "x", you group "+ 4" with the "x" and write it as a square:

-2(x + 4)² + 2 + 32

Then combine the numbers on the right side:

-2(x + 4)² + 34

Now, to do the translations:

1) Translating 4 units left. To do the horizontal translation, we do the opposite of what it says. Normally, moving 4 units left means we subtract 4. But we do the opposite, which is add 4. That makes it:

-2(x + 4 + 4)² + 34 or -2(x + 8)² + 34

2) Translating 3 units down, means we just subtract 3 from the overall so:

-2(x + 8)² + 34 - 3 or -2(x + 8)² + 31

So the vertex form is g(x) = -2(x + 8)² + 31

Standard form for a quadratic is: Ax² + Bx + C so to figure out what this is, we need to multiply this out

(x + 8)² = x² + 16x + 64 and multiplying that by -2 gives us -2x² - 32x - 128. Adding in + 31 gives:

The standard form is g(x) = -2x² -32x - 97

Description: This is an inverted parabola with a vertex at (-8, 31) [which we can read from the vertex form] with a vertical stretch factor of 2.

Answer 2

Hi Hadassah,

y=−2x^2−16x+2

Y = -2( x^2 +8x) +2 first factor of -2

Y = -2 [ (x + 4) ^2 -16] +2 in fact make a perfect square by   x^2 +8x

Y = -2 ( x + 4 )^2 +32 +2

Y = -2 ( x + 4 )^2 +34

4 units to the left 34 units to the up this parabola opens down vertex is x = -4

NOTE: you wrote 3 unit down this equation is 34 units up may be you make a mistake in writing

I hope it is useful,

Minoo