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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 23
Chapter 4, Problem 23

Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = (x - 2)2

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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 \([0, \infty)\).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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Properties 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.
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Domain & Range of Transformed Functions
Related Practice
Textbook Question

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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Textbook Question

Factor ƒ(x) into linear factors given that k is a zero. ƒ(x)=2x3−3x2−5x+6; k=1ƒ(x)=2x^3-3x^2-5x+6;^{\(\text{ }\)}k=1

Textbook Question

Use one of the end behavior diagrams below, to describe the end behavior of the graph of each polynomial function.

ƒ(x)=−4x3+3x2−1 ƒ(x)=-4x^3+3x^2-1

Textbook Question

Use synthetic division to perform each division. x7+1 / x+1

Textbook Question

Write each formula as an English phrase using the word varies or proportional. r = d/t, where r is the speed when traveling d miles in t hours.

Textbook Question

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