Question

I am confused on this problem and am looking for help. If anyone could help me it would be appreciated

Answer 1

Answer

$g(x)=-2(x+5)^{2}-3$

0. Horizontal translation 5 units.

,

1. Reflection over the x-axis

,

2. Vertical compression 2 units

,

3. Vertical translation down 3 units

$y=-(x+5)^2$

Explanation

• Writing the function in completed-square form.

As a ≠ 1, where a is the coefficient of the leading term, to write it in the completed-square form we have to make a = 1:

$g(x)=-2x^2-20x-53$$g(x)=-2(x^2+10x+\frac{53}{2})$

Now we have to take half of the x term and square it and add it to the function as follows:

$\frac{10}{2}^2=5^2=25$$g(x)=-2((x^2+10x+25)+\frac{53}{2}-25)$

Finally, we have a Perfect Squared Trinomial in the left side that we can rewrite as follows, obtaining our function g(x):

$g(x)=-2(x+5)^2+\frac{3\cdot-2}{2}$$g(x)=-2(x+5)^2-3$

As our parent function is:

$f(x)=x^2$

Then, the transformations that suffered were:

• Horizontal translation to the left 5 units

$y=(x+5)^2$

• Reflection over the x-axis

$y=-(x+5)^2$

• Vertical compression 2 units

$y=-2(x+5)^2$

• Vertical translation down 3 units

$g(x)=-2(x+5)^{2}-3$