What is the axis of symmetry, Vertex point, y intercept, and x intercept and the mirror of the y intercept of y=x2+6x+8
Help!! Change y-5= -2(x+3) into y=mx+b form:
in y = ax^2 + bx + c
the x of the vertex is -b/(2a)
for y = x^2 + 6x + 8
the x of the vertex is -6/2 = -3
and its y = 9 + 6(-3) + 8 = -1
so the vertex is (-3,-1)
the axis of symmetry is the vertical line passing through
the vertex, or y = -3
y-intercept, let x = 0 , so y = 8
x-intercept, let y = 0
x^2 + 6x + 8 = 0
(x+2)(x+4) = 0
x = -2, x = -4
(notice that the x of the vertex lies half way between the x-intercepts, that
would be another way to find the vertex.
You could also complete the square)
Don't know what you mean by the "mirror of the y-intercept"
We have to know who the mirror is, if it is the x-axis , then you would
have (0,-8).
I fail to see of what importance that would be.
2nd question: Change y-5= -2(x+3) into y=mx+b form
really?
Just expand the right side, then add 5 to both sides.
Simplify the right side
To find the axis of symmetry, vertex point, y-intercept, x-intercept, and the mirror of the y-intercept of the given function y = x^2 + 6x + 8, we can follow the steps below:
1. Axis of Symmetry:
The axis of symmetry is a vertical line that divides the parabola into two symmetric halves. It can be calculated using the formula: x = -b / (2a), where a and b are coefficients of the quadratic equation in standard form (i.e., ax^2 + bx + c).
In this case, the equation is y = x^2 + 6x + 8. Comparing it to the standard form, we have a = 1 and b = 6. Thus, the axis of symmetry can be determined as follows:
x = -(6) / (2 * 1) = -6 / 2 = -3.
Therefore, the axis of symmetry is x = -3.
2. Vertex Point:
The vertex point is the point where the parabola reaches its maximum or minimum value. To find it, we use the x-coordinate obtained from the axis of symmetry and substitute it into the original equation to determine the y-coordinate.
Using the x-coordinate of -3, we can find the y-coordinate:
y = (-3)^2 + 6(-3) + 8
y = 9 - 18 + 8
y = -1
So the vertex point is (-3, -1).
3. Y-intercept:
The y-intercept is the point where the parabola intersects the y-axis. To find it, we set x equal to 0 in the equation and solve for y.
Substituting x = 0 into the equation y = x^2 + 6x + 8:
y = (0)^2 + 6(0) + 8
y = 8
Thus, the y-intercept is (0, 8).
4. X-intercept:
The x-intercept(s) correspond to the point(s) where the parabola intersects the x-axis. To find it, we set y equal to 0 in the equation and solve for x.
Setting y = 0 in y = x^2 + 6x + 8:
x^2 + 6x + 8 = 0
To solve this quadratic equation, you can use factoring, completing the square, or the quadratic formula.
By factoring or using the quadratic formula, we find that the solutions are x = -2 and x = -4. Therefore, the x-intercepts are (-2, 0) and (-4, 0).
5. Mirror of Y-intercept:
To find the mirror of the y-intercept, you need to consider the line of symmetry (axis of symmetry). Since the y-intercept is a point on the parabola, its mirror image will also be equidistant from the line of symmetry.
The mirror image of the y-intercept (0, 8) with respect to the axis of symmetry (x = -3) will be the point on the opposite side of the parabola, at the same distance from the axis of symmetry. Therefore, we consider the x-coordinate of the y-intercept from the axis of symmetry, which is 3 units away.
So, the mirror of the y-intercept is (-6, 8).
In summary, for the equation y = x^2 + 6x + 8:
- Axis of symmetry: x = -3
- Vertex point: (-3, -1)
- Y-intercept: (0, 8)
- X-intercepts: (-2, 0) and (-4, 0)
- Mirror of the y-intercept: (-6, 8)