Ch. 7 - Conic Sections

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Blitzer 8th Edition

Ch. 7 - Conic Sections

문제 59

8장, 문제 59

In Exercises 57–62, use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function?
$y = - x^{2} + 4 x - 3$

검증된 단계별 안내

Identify the given quadratic function:

y = - x^{2} + 4 x - 3

Determine the direction in which the parabola opens by looking at the coefficient of

x^{2}

. Since it is negative (-1), the parabola opens downward.

Find the vertex of the parabola using the vertex formula for

x

= -\(\frac{b}{2a}\)

, where

a = -1

and

b = 4

. Calculate

x = -\(\frac{4}{2(-1)}\)

Substitute the

x

-value of the vertex back into the original equation to find the

y

-coordinate of the vertex, which gives the maximum value of

y

because the parabola opens downward.

Determine the domain and range: The domain of any quadratic function is all real numbers, so

\(\text{Domain}\) = (-\(\infty\), \(\infty\))

. The range is all

y

-values less than or equal to the vertex's

y

-coordinate, so

\(\text{Range}\) = (-\(\infty\), y_{vertex}]

. Since each

x

corresponds to exactly one

y

, the relation is a function.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Vertex of a Parabola

The vertex is the highest or lowest point on a parabola, found using the formula x = -b/(2a) for a quadratic y = ax^2 + bx + c. It helps identify the maximum or minimum value of the function, which is crucial for determining the range.

Direction of the Parabola

The direction a parabola opens depends on the coefficient 'a' in y = ax^2 + bx + c. If 'a' is positive, it opens upward; if negative, downward. This direction indicates whether the vertex is a maximum or minimum point, affecting the range of the relation.

Domain and Range of Quadratic Functions

The domain of a quadratic function is all real numbers since x can take any value. The range depends on the vertex and the parabola's direction: if it opens downward, the range is all y-values less than or equal to the vertex's y-coordinate; if upward, all y-values greater than or equal to it.

In Exercises 57–62, use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? y=−x2+4x−3y=-x^2+4x-3