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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 25
Chapter 4, Problem 25

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. f(x)=4−(x−1)2

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Identify the given quadratic function: \(f(x) = 4 - (x - 1)^2\). Notice it is in vertex form \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
Determine the vertex by comparing: here, \(h = 1\) and \(k = 4\), so the vertex is at the point \((1, 4)\).
Find the axis of symmetry, which is the vertical line passing through the vertex's x-coordinate: \(x = 1\).
Calculate the y-intercept by evaluating \(f(0)\): substitute \(x = 0\) into the function to find the point where the graph crosses the y-axis.
Determine the domain and range: the domain of any quadratic function is all real numbers, \((-\infty, \infty)\). Since the parabola opens downward (because of the negative sign before the squared term), the range is all \(y\) values less than or equal to the vertex's y-value, so \((-\infty, 4]\).

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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, represented by a parabola. For functions in the form f(x) = a(x - h)^2 + k, the vertex is at (h, k). It helps determine the shape and position of the parabola and is essential for sketching the graph.
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Vertex Form

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. Its equation is x = h, where h is the x-coordinate of the vertex. This line helps in graphing and understanding the parabola's symmetry.
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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 downward, the range is all values less than or equal to the vertex's y-coordinate, and if upward, all values greater than or equal to it.
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Domain & Range of Transformed Functions
Related Practice
Textbook Question

In Exercises 25–26, graph each polynomial function. f(x)=2x2(x−1)3(x+2)f(x) = 2x^2(x - 1)^3(x + 2)

Textbook Question

Divide using synthetic division. (x2−5x−5x3+x4)÷(5+x)

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. x2 ≤ 4x − 2

Textbook Question

In Exercises 25–26, graph each polynomial function. f(x)=−x3(x+4)2(x−1)f(x) = -x^3(x + 4)^2(x-1)

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Textbook Question

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=2(x−5)(x+4)2

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. h(x)=x/x(x+4)