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For each graph, identify the domain, range, x-intercepts, y-intercepts, relative extrema, absolute extrema, intervals of increasing/decreasing, and concavity.

For each graph, identify the domain, range, x-intercepts, y-intercepts, relative extrema-example-1
User IndoKnight
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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).

User Lesleyann
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