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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 15
Chapter 4, Problem 15

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−x2−2x+8

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Identify the quadratic function given: \(f(x) = -x^2 - 2x + 8\). This is in the standard form \(f(x) = ax^2 + bx + c\) where \(a = -1\), \(b = -2\), and \(c = 8\).
Recall that the vertex of a parabola defined by \(f(x) = ax^2 + bx + c\) has an \(x\)-coordinate given by the formula \(x = -\frac{b}{2a}\).
Substitute the values of \(a\) and \(b\) into the vertex formula: \(x = -\frac{-2}{2 \times -1}\).
Simplify the expression to find the \(x\)-coordinate of the vertex.
To find the \(y\)-coordinate of the vertex, substitute the \(x\)-value back into the original function: \(f(x) = -x^2 - 2x + 8\). Calculate \(f(\text{x-coordinate})\) to get the \(y\)-coordinate.

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Key Concepts

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

Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as f(x) = ax^2 + bx + c. Its graph is a parabola, which can open upwards or downwards depending on the sign of the coefficient a.
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Solving Quadratic Equations Using The Quadratic Formula

Vertex of a Parabola

The vertex is the highest or lowest point on the parabola, representing its maximum or minimum value. For f(x) = ax^2 + bx + c, the vertex's x-coordinate is found using -b/(2a), and the y-coordinate is f(-b/(2a)).
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Horizontal Parabolas

Evaluating Functions

Evaluating a function means substituting a specific value for the variable and calculating the result. To find the vertex's y-coordinate, substitute the x-value of the vertex into the original quadratic function.
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Evaluating Composed Functions
Related Practice
Textbook Question

In Exercises 15–18, 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).] <IMAGE>

f(x)=−x4+x2f(x)=−x^4+x^2

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)=2x3+x2−3x+1

Textbook Question

Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square of w.

Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As x→1+, f(x)→x\(\to\)1^{+},\(\text{ }\)f(x)\(\to\) __

Textbook Question

Divide using long division. State the quotient, and the remainder, r(x).2x5−8x4+2x3+x22x3+1\(\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}\)

Textbook Question

Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square root of w.