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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 9
Chapter 4, Problem 9

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2(x−3)2+1

Verified step by step guidance
1
Identify the form of the quadratic function. The given function is in vertex form: \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
Compare the given function \(f(x) = 2(x - 3)^2 + 1\) to the vertex form to find \(h\) and \(k\). Here, \(h = 3\) and \(k = 1\).
Recall that the vertex of the parabola is at the point \((h, k)\), so the vertex coordinates are \((3, 1)\).
Understand that since the coefficient \(a = 2\) is positive, the parabola opens upwards, confirming the vertex is a minimum point.
Summarize that the vertex coordinates for the parabola defined by \(f(x) = 2(x - 3)^2 + 1\) are \((3, 1)\).

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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 expressed as f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. This form makes it easy to identify the vertex directly without completing the square or using calculus.
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Vertex Form

Coordinates of the Vertex

The vertex of a parabola given in vertex form f(x) = a(x - h)^2 + k is the point (h, k). This point is either the maximum or minimum of the function depending on the sign of 'a'. For the function f(x) = 2(x - 3)^2 + 1, the vertex is at (3, 1).
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Graphs and Coordinates - Example

Effect of the Coefficient 'a' on the Parabola

The coefficient 'a' in the quadratic function affects the parabola's width and direction. If 'a' is positive, the parabola opens upward, indicating a minimum vertex; if negative, it opens downward, indicating a maximum vertex. The larger the absolute value of 'a', the narrower the parabola.
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Related Practice
Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.


As x→−3−x\(\to\)-3^{-}, f(x)→f\(\left\)(x\(\right\))\(\to\)____

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Textbook Question

Determine which functions are polynomial functions. For those that are, identify the degree. f(x)=(x2+7)/x3f(x)=(x^2+7)/x^3

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Textbook Question

Determine which functions are polynomial functions. For those that are, identify the degree. f(x)=(x2+7)/3f(x)=(x^2+7)/3

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Textbook Question

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x3+x2−4x−4

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Textbook Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] f(x)=−x3+x2+2xf(x) = -x^3 + x^2 + 2x

a.b. c. d.

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Textbook Question

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as a and b and inversely as the square root of c. y = 12 when a = 3, b = 2, and c = 25. Find y when a = 5, b = 3 and c = 9.

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