Domain: The domain is all real numbers (-∞, ∞).
Range: The range is (-∞, -2]).
X-intercepts: it has no x-intercept.
Y-intercept: The point is (0, -4).
Relative Extrema is (1,-2)
Absolute Extrema: There is no absolute maximum but the vertex is the absolute minimum.
Intervals of Increasing/Decreasing: The parabola is decreasing for x < 1 and increasing for x > 1.
Concavity: The parabola is concave down.
How to analyze a parabola.
For a parabola in the form y = a(x - h)² + k
where
(h, k) is the vertex, (1, -2) we can write the equation as:
y = a(x - 1)² - 2
y = a(0 - 1)² - 2
y = a - 2
Since the y-intercept is -4,
set y = -4 and solve for a.
-4 = a - 2
Solving for a, we get a = -2.
The equation of the parabola is
y = -2(x - 1)² - 2
let's analyze the properties you mentioned:
Domain: The domain is all real numbers (-∞, ∞).
The curve extends infinitely on both side of x axis.
Range:Since the parabola opens downward, the range is (-∞, -2]). It opens downward.
X-intercepts: There is no x-intercept, it has no real roots.
Y-intercept: Given as -4, the y-intercept is at the point (0, -4).
Relative Extrema: Since the parabola opens downward, the vertex (1, -2) is the relative maximum.
Absolute Extrema: There is no absolute maximum (since it extends downward indefinitely), but the vertex is the absolute minimum.
Intervals of Increasing/Decreasing: The parabola is decreasing for x < 1 and increasing for x > 1.
Concavity: The parabola is concave down (opens downward).