교과서 질문
Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=4x7-x5+x3-1
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Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
4장, 문제 23
Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = (x - 2)2
검증된 단계별 안내1
Identify the given quadratic function: \(f(x) = (x - 2)^2\). This is in vertex form, which is \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
Find the vertex by comparing \(f(x) = (x - 2)^2\) to the vertex form. Here, \(h = 2\) and \(k = 0\), so the vertex is at \((2, 0)\).
Determine the axis of symmetry, which is the vertical line that passes through the vertex. The axis is \(x = 2\).
State the domain of the function. Since this is a quadratic function, the domain is all real numbers, written as \((-\infty, \infty)\).
Find the range by considering the vertex and the direction the parabola opens. Since \(a = 1 > 0\), the parabola opens upward, so the range is \([0, \infty)\).
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Vertex of a Quadratic Function
The vertex is the highest or lowest point on the graph of a quadratic function, representing its maximum or minimum value. For functions in the form f(x) = (x - h)^2 + k, the vertex is at (h, k). In this question, the vertex is at (2, 0), indicating the parabola opens upward from this point.
추천 영상:
08:07
Vertex Form
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex and has the equation x = h for a quadratic in vertex form. Here, the axis of symmetry is x = 2, reflecting the parabola's symmetry about this line.
추천 영상:
07:42Properties of Parabolas
Domain and Range of Quadratic Functions
The domain of any quadratic function is all real numbers since x can take any value. The range depends on the vertex and the parabola's direction; for f(x) = (x - 2)^2, the parabola opens upward with a minimum at y = 0, so the range is y ≥ 0.
추천 영상:
4:22Domain & Range of Transformed Functions
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Solve each polynomial inequality. Give the solution set in interval notation.
(a) -x(x - 1)(x - 2) ≥ 0
(b) -x(x - 1)(x - 2) > 0
(c) -x(x - 1)(x - 2) ≤ 0
(d) -x(x - 1)(x - 2) < 0
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