Please help, I am really having trouble understanding this problem.
what are the foci of the hyperbola given by the equation 16y^2-9x^2=144
Of course, I can help you with that problem. To determine the foci of the hyperbola given by the equation 16y^2 - 9x^2 = 144, we need to identify the standard form of the hyperbola equation, which is:
(x-h)^2/a^2 - (y-k)^2/b^2 = 1 if the hyperbola is horizontal (a^2 > b^2),
or
(y-k)^2/a^2 - (x-h)^2/b^2 = 1 if the hyperbola is vertical (b^2 > a^2).
Let's rearrange the given equation to match the standard form.
16y^2 - 9x^2 = 144
First, divide the equation by 144 to make the right-hand side equal to 1:
(y^2/9) - (x^2/16) = 1
Since the coefficient of y^2 is positive and greater than the coefficient of x^2, we know that this hyperbola is vertical. So, we can rewrite the equation as:
(y-0)^2/9 - (x-0)^2/16 = 1
From this, we can identify the center of the hyperbola as (h, k) = (0, 0).
Next, we can find the values of a^2 and b^2. In this case, a^2 = 16 and b^2 = 9.
To determine the foci, we can use the formula c^2 = a^2 + b^2, where c is the distance between the center and the foci.
Substituting the values, we have:
c^2 = 16 + 9
c^2 = 25
Taking the square root of both sides, we find:
c = ±5
Since the hyperbola is vertical, the foci are located at (h, k ± c). Therefore, the foci of the hyperbola are (0, 5) and (0, -5).
***I USED A DIFFERENT PROBLEM. FOLLOW THE STEPS!!!
The vertex is either the minimum point of the graph of the maximum point of the graph if the equation is a quadratic equation.
if it is the minimum point on the graph, than any other point on the graph will be above it.
if it is the maximum point on the graph, than any other point on the graph will be below it.
(-3,125) is above (0,8).
(0,8) is therefore a minimum point on the graph and the graph opens up.
in fact, your equation is y = 13x^2 + 8.
when x = 0, y = 8 as you can see by replacing x with 0 and getting 13*0^2 + 8 = 8 which becomes 8 = 8.
when x = -3, your equation becomes y = 13 * (-3)^2 + 8 which becomes y = 13 * 9 + 8 which becomes y = 117 + 8 which becomes y = 125.
when x = -3, y = 125 which corresponds to the REQUIREMENTS that the point (-3,125) be on the grpah of the equation.
your equation is shown below:
graph%28600%2C600%2C-4%2C4%2C-5%2C200%2C13x%5E2+%2B+8%2C8%2C125%29
i drew horizontal lines at y = 8 and y = 125.
the intersection of those lines with the graph gives you the value of x when you draw a vertical line from the intersection to the x-axis.