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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 21
Chapter 4, Problem 21

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. y−1=(x−3)2

Verified step by step guidance
1
Rewrite the given equation in standard vertex form: \(y - 1 = (x - 3)^2\). This shows the vertex is at the point \((3, 1)\).
Identify the axis of symmetry, which is the vertical line passing through the vertex's x-coordinate. So, the axis of symmetry is \(x = 3\).
Find the y-intercept by substituting \(x = 0\) into the equation: \(y - 1 = (0 - 3)^2\), then solve for \(y\).
Find the x-intercepts by setting \(y = 0\) and solving the equation \(0 - 1 = (x - 3)^2\) for \(x\).
Determine the domain and range: The domain of any quadratic function is all real numbers, \((-\infty, \infty)\). Since the parabola opens upward and the vertex is the minimum point at \(y=1\), the range is \([1, \infty)\).

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Key Concepts

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

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex, which is the highest or lowest point on the graph, depending on the parabola's orientation.
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Vertex Form

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. For a quadratic in vertex form y = a(x - h)^2 + k, the axis of symmetry is the line x = h.
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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: if it opens upward, the range is y ≥ k; if downward, y ≤ k, where k is the y-coordinate of the vertex.
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Domain & Range of Transformed Functions
Related Practice
Textbook Question

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. 6x3+25x2−24x+5=0

Textbook Question

Divide using synthetic division. (4x3−3x2+3x−1)÷(x−1)

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Textbook Question

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. f(x)=x/(x+4)

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Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 2x2+3x>02x^2+3x>0

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Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=5x4+7x2−x+9f(x)=5x^4+7x^2−x+9

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Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 3x2 − 5x ≤ 0

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