Give vertex, axis, x-intercepts and y-intercepts to parabola
y= -3x^2-6x+4
Y = -3X^2 - 6X + 4.
Vertex Form: y = a(x - h)^2 + k.
h = Xv = -b/2a = 6/-6 = -1.
Substitute -1 for x in the given Eq:
k = -3(-1)^2 - 6(-1) + 4=3 + 6 + 4 = 7.
V(h , k) = V(-1 , 7).
Vertex Form: y = -3(x + 1)^2 + 7.
Axis = h =Xv = -1.
The axis is a vertical line and equals
-1 for all values of y.
Let x = 0 and solve for y:
y = 3*0^2 - 6*0 + 4 = 4 = Y-Int.
The solutions,x-Intercepts, and roots
are all the same.
x = (6 +- sqrt(6^2 + 48)) / -6,
x = (6 +- sqrt(84)) / -6,
x = (6 +- 9.17) / -6,
x = (6 + 9.17) /-6 = - 2.5275.
x = (6 - 9.17) / -6 = 0.5275.
Solution set: x = -2.5275, x = 0.5275.
To find the vertex, axis, x-intercepts, and y-intercept of the parabola y = -3x^2 - 6x + 4, we can use the equation of the parabola in the general form:
y = ax^2 + bx + c
Comparing this equation with the given equation, we can determine the values of a, b, and c:
a = -3
b = -6
c = 4
1. Vertex:
The vertex of a parabola can be found using the formula x = -b / (2a) and then substituting the value of x into the equation to calculate the y-coordinate. Let's plug in the values:
x = -(-6) / (2 * -3)
x = 6 / -6
x = -1
Now substitute x = -1 into the equation to find the y-coordinate:
y = -3(-1)^2 - 6(-1) + 4
y = -3 + 6 + 4
y = 7
Therefore, the vertex is (-1, 7).
2. Axis:
The axis of symmetry is a vertical line that passes through the vertex. Its equation is simply x = x-coordinate of the vertex. In this case, the axis of symmetry is x = -1.
3. X-intercepts:
To find the x-intercepts of the parabola, we need to solve the equation y = 0 for x. Let's set y = 0:
-3x^2 - 6x + 4 = 0
There are a few ways to solve this quadratic equation, but let's use the quadratic formula. The quadratic formula states that for any equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying this to our equation, we have:
x = (-(-6) ± √((-6)^2 - 4(-3)(4))) / (2(-3))
x = (6 ± √(36 + 48)) / (-6)
x = (6 ± √84) / (-6)
x = (6 ± 2√21) / (-6)
Hence, the x-intercepts are (1 - 2√21, 0) and (1 + 2√21, 0).
4. Y-intercept:
The y-intercept occurs when x = 0. To find it, we can substitute x = 0 into the equation:
y = -3(0)^2 - 6(0) + 4
y = 4
Therefore, the y-intercept is (0, 4).
To summarize:
Vertex: (-1, 7)
Axis: x = -1
X-intercepts: (1 - 2√21, 0) and (1 + 2√21, 0)
Y-intercept: (0, 4)