y^2= 9(x^2)+20x+3 how to identify its orientation and the locaation of its vertices?
sqrt(x^4+8x^3)=sqrt(9x^2+20x+3) How to identify the parabola's orientation? up or down?
thank you
To determine the orientation of a quadratic equation, such as y^2 = 9(x^2) + 20x + 3, we need to analyze the coefficient of the x^2 term. Let's break down the steps:
1. Rewrite the equation in the standard form by isolating the quadratic term on one side:
y^2 - 9x^2 - 20x - 3 = 0
2. Compare the coefficient of the x^2 term to determine if it is positive or negative:
In this case, the coefficient of the x^2 term is -9. Since it is negative, the parabola opens downwards.
3. Therefore, the orientation of the parabola described by the equation y^2 = 9(x^2) + 20x + 3 is downwards.
Now, let's move on to identifying the location of its vertices.
1. The vertex of a parabola in the form y^2 = 4ax can be found by using the formula h = -b/2a, where h is the x-coordinate of the vertex.
2. In this case, the equation is y^2 = 9(x^2) + 20x + 3. Comparing it to y^2 = 4ax, we can determine that a = 9 and b = 20.
3. Plug the values of a and b into the formula h = -b/2a:
h = -20 / (2 * 9) = -20 / 18 = -10 / 9
4. The x-coordinate of the vertex is -10/9, which determines its horizontal position.
5. To find the y-coordinate, substitute the x-coordinate back into the original equation:
y^2 = 9((-10/9)^2) + 20(-10/9) + 3
6. Compute the value to find the y-coordinate.