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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 30
Chapter 4, Problem 30

Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = x2 + 6x + 5

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Identify the quadratic function given: \(f(x) = x^2 + 6x + 5\).
Rewrite the quadratic function in vertex form by completing the square: Start with \(f(x) = x^2 + 6x + 5\). To complete the square, take half of the coefficient of \(x\) (which is 6), divide by 2 to get 3, then square it to get 9. Add and subtract 9 inside the function to maintain equality.
Express the function as \(f(x) = (x^2 + 6x + 9) + 5 - 9\), which simplifies to \(f(x) = (x + 3)^2 - 4\). This form reveals the vertex clearly.
From the vertex form \(f(x) = (x + 3)^2 - 4\), identify the vertex as \((-3, -4)\) and the axis of symmetry as the vertical line \(x = -3\).
Determine the domain and range: The domain of any quadratic function is all real numbers, so \((-\infty, \infty)\). Since the parabola opens upward (coefficient of \(x^2\) is positive), the range is \([-4, \infty)\), starting from the vertex's \(y\)-value.

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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, representing its maximum or minimum value. It can be found using the formula (-b/2a, f(-b/2a)) when the function is in standard form f(x) = ax^2 + bx + c. The vertex helps in graphing and understanding the function's behavior.
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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 = -b/(2a) for a quadratic function in standard form. This line is crucial for graphing and analyzing the symmetry of the parabola.
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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: if the parabola opens upward (a > 0), the range is all values greater than or equal to the vertex's y-coordinate; if it opens downward (a < 0), the range is all values less than or equal to the vertex's y-coordinate.
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Domain & Range of Transformed Functions
Related Practice
Textbook Question

Resistance of a Wire The resistance in ohms of a platinum wire temperature sensor varies directly as the temperature in kelvins (K). If the resistance is 646 ohms at a temperature of 190 K, find the resistance at a temperature of 250 K.

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=-2x(x-3)(x+2)

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Textbook Question

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ƒ(x)=(x+7)(x+1)ƒ(x)=\(\frac{(x+7)}{(x+1)}\)

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Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation. (x - 3)(x - 4)(x - 5)2 ≤ 0

Textbook Question

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Textbook Question

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. ƒ(x)=1/(x+4)