Textbook Question
Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(x2-2x-3)/(2x2-x-10)
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 45
Several graphs of the quadratic function ƒ(x) = ax2 + bx + c are shown below. For the given restrictions on a, b, and c, select the corresponding graph from choices A–F. (Hint: Use the discriminant.) a < 0; b2 - 4ac = 0
Verified step by step guidance1
Recall the quadratic function is given by \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
Note the given conditions: \(a < 0\) and the discriminant \(b^2 - 4ac = 0\). The discriminant tells us about the nature of the roots of the quadratic.
Since \(b^2 - 4ac = 0\), the quadratic has exactly one real root, meaning the graph touches the x-axis at exactly one point (a repeated root).
Because \(a < 0\), the parabola opens downward, so the vertex is a maximum point and the graph is concave down.
To select the correct graph, look for the parabola that opens downward and just touches the x-axis at one point (the vertex), reflecting the repeated root condition.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Function and Its Graph
A quadratic function is a polynomial of degree two, expressed as ƒ(x) = ax² + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. The coefficients a, b, and c determine the shape and position of the parabola on the coordinate plane.
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Graphs of Logarithmic Functions
Discriminant of a Quadratic Equation
The discriminant, given by b² - 4ac, indicates the nature of the roots of a quadratic equation. If the discriminant equals zero, the quadratic has exactly one real root (a repeated root), meaning the parabola touches the x-axis at a single point (vertex).
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Effect of Coefficient 'a' on Parabola Orientation
The sign of the coefficient 'a' determines the direction the parabola opens. If a < 0, the parabola opens downward, forming a maximum point at the vertex. This affects the graph's shape and helps identify the correct graph when combined with other conditions.
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Related Practice
Textbook Question
Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x4+x3-6x2-7x-2
Textbook Question
If the given term is the dominating term of a polynomial function, what can we conclude about each of the following features of the graph of the function? (a) domain (b) range (c) end behavior (d) number of zeros (e) number of turning points 10x7
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Textbook Question
Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = -3x2 + 18x + 1
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Textbook Question
Solve each polynomial inequality. Give the solution set in interval notation. x5 + x2 + 2 ≥ x4 + x3 + 2x
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