The equation of the parabola ( y = 4x^2 ) can be rewritten in the standard form ( y = 4p(x - h)^2 + k ), where ( (h, k) ) is the vertex. Here, it is clear that the vertex is at the origin (0, 0) and ( 4p = 4 ), giving ( p = 1 ). The focus of the parabola is located at ( (h, k + p) ), so the focus is at the point ( (0, 1) ).
The equation ( y = 4x^2 ) represents a parabola that opens upwards. To find the focus, we can rewrite it in the standard form ( y = 4p(x - h)^2 + k ), where ( (h, k) ) is the vertex and ( p ) is the distance from the vertex to the focus. Here, the vertex is at the origin ( (0, 0) ) and ( 4p = 4 ), so ( p = 1 ). Thus, the focus of the parabola is located at the point ( (0, 1) ).
yes!
The equation y = 4x^2 + 5 is a parabola
It is a parabola with its vertex at the origin and the arms going upwards.
The focus of a parabola is a specific point that defines its shape, while the directrix is a line used in the definition of a parabola. If the directrix is given as ( y = -2 ), the parabola opens either upwards or downwards. The focus would be located at a point above or below this directrix, depending on the orientation of the parabola. Specifically, for a parabola that opens upwards, the focus would be positioned at ( (h, k + p) ), where ( p ) is the distance from the vertex to the focus, and the vertex would be located at ( (h, -2 + p) ).
Their noses are both at the origin, and they both open upward, but y=4x2 is a much skinnier parabola.
The equation ( y = 4x^2 ) represents a parabola that opens upwards. To find the focus, we can rewrite it in the standard form ( y = 4p(x - h)^2 + k ), where ( (h, k) ) is the vertex and ( p ) is the distance from the vertex to the focus. Here, the vertex is at the origin ( (0, 0) ) and ( 4p = 4 ), so ( p = 1 ). Thus, the focus of the parabola is located at the point ( (0, 1) ).
yes!
I think you are talking about the x-intercepts. You can find the zeros of the equation of the parabola y=ax2 +bx+c by setting y equal to 0 and finding the corresponding x values. These will be the "roots" of the parabola.
The equation y = 4x^2 + 5 is a parabola
It is a parabola with its vertex at the origin and the arms going upwards.
The focus of a parabola is a fixed point that lies on the axis of the parabola "p" units from the vertex. It can be found by the parabola equations in standard form: (x-h)^2=4p(y-k) or (y-k)^2=4p(x-h) depending on the shape of the parabola. The vertex is defined by (h,k). Solve for p and count that many units from the vertex in the direction away from the directrix. (your focus should be inside the curve of your parabola)
The focus of a parabola is a specific point that defines its shape, while the directrix is a line used in the definition of a parabola. If the directrix is given as ( y = -2 ), the parabola opens either upwards or downwards. The focus would be located at a point above or below this directrix, depending on the orientation of the parabola. Specifically, for a parabola that opens upwards, the focus would be positioned at ( (h, k + p) ), where ( p ) is the distance from the vertex to the focus, and the vertex would be located at ( (h, -2 + p) ).
The primary focal chord of a parabola is a line segment that passes through the focus of the parabola and has its endpoints on the parabola itself. For a standard parabola defined by the equation (y^2 = 4px), the focus is located at the point ((p, 0)). The primary focal chord is unique in that it is perpendicular to the axis of symmetry of the parabola and is the longest chord that can be drawn through the focus.
To write the equation of a parabola with its vertex at the origin (0, 0) and a focus at (0, 60), you first identify the orientation of the parabola. Since the focus is above the vertex, the parabola opens upwards. The standard form of the equation for a parabola that opens upwards is ( y = \frac{1}{4p}x^2 ), where ( p ) is the distance from the vertex to the focus. Here, ( p = 60 ), so the equation becomes ( y = \frac{1}{240}x^2 ).
x2 = 16y The standard formula for a parabola with its vertex at the origin (0, 0) and a given focus (and the y-axis as an axis of symmetry) is as follows: x2 = 4cy In this case, the c is the y value of the focus. The focus in this case was (0, 4), and the y value in the focus is 4. That makes the c = 4. Further, that makes the equation for this parabola x2 = 4 (c)y = 4 (4)y = 16y Given that the vertex was the origin, (0, 0), and the focus is (0, 4), we can conclude that the axis of symmetry is the y-axis because the y value of the focus is 0. We can also conclude that the parabola opens up, because the focus has a positive y value.
A sideways parabola is commonly referred to as a "horizontal parabola." Unlike the standard vertical parabola, which opens upwards or downwards, a horizontal parabola opens to the left or right. Its general equation takes the form (y^2 = 4px) for a right-opening parabola or (y^2 = -4px) for a left-opening parabola, where (p) determines the distance from the vertex to the focus.