1.) Y=-lx+1l+3
Are these correct?
y-intercept=(0,2)
x-intercept=(4,0) (-2,0)
vertex=(-1,3)
domain=all real numbers
range: y<=3
2.) A parabola has vertex (2,3) and it contains (0,0)....find the equation of the sleeping parabola.
#1, ok
#2, "sleeping parabola" ?????
1.) To check if the given values are correct, we can substitute them into the equation and see if they satisfy it.
For the y-intercept, you are given that the point is (0,2). To check if it's correct, substitute x=0 into the equation:
Y = |-0 + 1| + 3
Y = 1 + 3
Y = 4
The y-coordinate of the y-intercept is 4, not 2. Therefore, the y-intercept is not correct.
Next, for the x-intercepts, you are given two points: (4,0) and (-2,0). Substitute these values into the equation:
For (4,0):
Y = |-4 + 1| + 3
Y = 3 + 3
Y = 6
For (-2,0):
Y = |-(-2) + 1| + 3
Y = 3 + 3
Y = 6
The y-coordinate should be 0 for both x-intercepts, not 6. Therefore, the x-intercepts you provided are not correct.
Moving on to the vertex, you are given (-1,3). Substitute these values into the equation:
Y = |-(-1) + 1| + 3
Y = 1 + 3
Y = 4
The y-coordinate of the vertex is indeed 3, so the given vertex is correct.
To determine the domain, note that this is an absolute value function, and its domain is all real numbers. Therefore, the given domain is correct.
Lastly, for the range, you are given y ≤ 3. This means that all y-values must be less than or equal to 3. Substituting various y-values greater than 3, we find that they do not satisfy the given equation. Therefore, the given range is correct.
In summary, the correct values are as follows:
- Y-intercept: (0,4)
- X-intercepts: None
- Vertex: (-1,3)
- Domain: All real numbers
- Range: y ≤ 3
2.) To find the equation of a parabola with a vertex and a point on it, we need to use the standard form of the equation:
y = a(x - h)^2 + k
where (h, k) denotes the vertex.
Given that the vertex is (2, 3) and it contains the point (0, 0), substitute these values into the equation:
0 = a(0 - 2)^2 + 3
0 = a(-2)^2 + 3
0 = 4a + 3
Solving this equation for 'a,' we get:
4a = -3
a = -3/4
Therefore, the equation of the sleeping parabola is:
y = (-3/4)(x - 2)^2 + 3