-x^2 - 8x + y - 18 = 0
Please Help, I don't understand problems like this, Please show steps.
Please Help, I don't understand problems like this, Please show steps.
-
Notice that we have two variants for the standard equation of a parabola:
y - k = [1/(4p)](x - h)^2 OR x - h = [1/(4p)](y - k)^2.
In order to make our solution, notice that the first flavor has a x^2 term, while the second has a y^2 term. The equation given has a x^2 term, so we need to use:
y - k = [1/(4p)](x - h)^2.
A parabola in this form has its vertex at (h, k), its focus |p| units directly above the vertex for p > 0 (if p < 0, then the focus is below the vertex instead), and its directix is a horizontal line |p| units below the vertex for p > 0 (if p < 0, then the directix lies above the vertex instead).
Now, we need to put the given equation into this form. To do this, notice that this form has all terms involving y on one side and all terms involving x on the other. Doing this with the given equation yields:
y - 18 = x^2 + 8x.
At this point, we need to complete the square on the right side. Recall that in order to complete the square, we need to take the x coefficient, halve it, and then square the result. Here, the x coefficient is 8, which halves to get 4, and squares to get 16. So, adding 16 to both sides gives:
y - 18 + 16 = x^2 + 8x + 16 ==> y - 2 = (x + 4)^2.
Note that the right side is a perfect square binomial and factored as such.
Now, the 1/(4p) term is just the number outside on the right side. There is an understood 1 on the right side since:
(x + 4)^2 = 1(x + 4)^2.
Then, we see that:
1/(4p) = 1 ==> p = 1/4.
So, the given equation in standard form is:
y - 2 = [1/)4 * 1/4)](x + 4)^2.
Comparing this to the standard equation yields the vertex to be (-4, 2). Then, since p = 1/4, the focus lies 1/4 units directly above the vertex at (-4, 2 + 1/4) = (-4, 9/4), and the directix is a horizontal line 1/4 units below the vertex---this has equation y = 2 - 1/4 = 7/4.
I hope this helps!
y - k = [1/(4p)](x - h)^2 OR x - h = [1/(4p)](y - k)^2.
In order to make our solution, notice that the first flavor has a x^2 term, while the second has a y^2 term. The equation given has a x^2 term, so we need to use:
y - k = [1/(4p)](x - h)^2.
A parabola in this form has its vertex at (h, k), its focus |p| units directly above the vertex for p > 0 (if p < 0, then the focus is below the vertex instead), and its directix is a horizontal line |p| units below the vertex for p > 0 (if p < 0, then the directix lies above the vertex instead).
Now, we need to put the given equation into this form. To do this, notice that this form has all terms involving y on one side and all terms involving x on the other. Doing this with the given equation yields:
y - 18 = x^2 + 8x.
At this point, we need to complete the square on the right side. Recall that in order to complete the square, we need to take the x coefficient, halve it, and then square the result. Here, the x coefficient is 8, which halves to get 4, and squares to get 16. So, adding 16 to both sides gives:
y - 18 + 16 = x^2 + 8x + 16 ==> y - 2 = (x + 4)^2.
Note that the right side is a perfect square binomial and factored as such.
Now, the 1/(4p) term is just the number outside on the right side. There is an understood 1 on the right side since:
(x + 4)^2 = 1(x + 4)^2.
Then, we see that:
1/(4p) = 1 ==> p = 1/4.
So, the given equation in standard form is:
y - 2 = [1/)4 * 1/4)](x + 4)^2.
Comparing this to the standard equation yields the vertex to be (-4, 2). Then, since p = 1/4, the focus lies 1/4 units directly above the vertex at (-4, 2 + 1/4) = (-4, 9/4), and the directix is a horizontal line 1/4 units below the vertex---this has equation y = 2 - 1/4 = 7/4.
I hope this helps!