PLZZZZZ HELP 100 POINTS
2 answers:
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Answer:
The solutions are (-2 , 0) and (1 , 3)
Step-by-step explanation:
∵ -x² + 4 = x + 2
* y = -x² + 4 ⇒ is quadratic represented by parabola
The parabola open downward because coefficient of x² is negative
The x-coordinate of its vertex = -b/2a, where b is the
coefficient of x and a is the coefficient of x²
∴ x = 0/2(-1) = 0
∴ The y-coordinate of the vertex = (0)² + 4 = 4
∴ The maximum point of the parabola is (0 , 4)
∵ -x² + 4 intersects x-axis at y = 0
∴ -x² + 4 = 0 ⇒ -x² = -4 ⇒ x² = 4
∵ x² = 4
∴ x = ±√4 = ± 2
∴ the parabola intersects x-axis at -2 , 2
∵ y = x + 2 represented by a line its slope = 1
It intersects y-axis at 2
∵ x + 2 intersects x-axis at y = 0
∴ x + 2 = 0
∴ x = -2
∴ The parabola and the line intersect each other at
x = -2 and y = 0
To find all the point of intersection between the 2 equations we will solve them as a system of equations
∵ y= -x² + 4 and y = x + 2
∴ -x² + 4 = x + 2
∴ -x² + 4 - x - 2 = 0
∴ -x² - x + 2 = 0 ⇒ × -1
∴ x² + x - 2 = 0 ⇒ factorize
∴ (x + 2)(x - 1) = 0
∴ x + 2 = 0 ⇒ x = -2
∴ x - 1 = 0 ⇒ x = 1
* when x = -2 ⇒ y = -2 + 2 = 0 ⇒ (-2 , 0)
* when x = 1 ⇒ y = 1 + 2 = 3 ⇒ (1 , 3)
The solution graphically
Answer:
- p = -3
- v: (4, -1)
- f: (4, -4)
- d: y = 2
Step-by-step explanation:
The equation ...
(x -h)² = 4p(y -k)
is the equation of a parabola with vertex (h, k) and vertex-focus distance of p. When p is negative (as here), it means the focus is below the vertex, and the parabola opens downward. The directrix is p units on the other side of the vertex from the focus.
The vertex for equation ...
(x -4)² = -12(y +1)
is (h, k) = (4, -1). The value of p is (-12/4) = -3, so the focus is (4, -1-3) = (4, -4), and the directrix is y = -1+3 = 2.
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<em>Additional comment</em>
As a check on the graph, you can remember that any point on the parabola is equidistant from the focus and directrix. You will notice the vertex is halfway between the focus and directrix, so you know that relationship is true there.
You will also notice that a horizontal line from the focus to the parabola intersects the same distance away from the focus as from the directrix. Thus the parabola distance relation holds at that point as well.
