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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 47
Chapter 4, Problem 47

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

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1
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 determines the nature of the roots of the quadratic equation.
Since \(a < 0\), the parabola opens downward, meaning the graph is concave down.
Because the discriminant \(b^2 - 4ac < 0\), the quadratic has no real roots, so the graph does not intersect the x-axis.
To select the correct graph, look for a downward-opening parabola that lies entirely above or below the x-axis without crossing it.

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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 f(x) = ax^2 + 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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Discriminant of a Quadratic Equation

The discriminant, given by b^2 - 4ac, helps determine the nature of the roots of a quadratic equation. If the discriminant is less than zero, the quadratic has no real roots and the parabola does not intersect the x-axis. This information is crucial for identifying the correct graph.
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Effect of Coefficients on the Parabola

The coefficient a controls the direction and width of the parabola, with a < 0 indicating it opens downward. The values of b and c affect the vertex's position and the parabola's vertical shift. Understanding these effects helps match the quadratic function to its graph under given restrictions.
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Related Practice
Textbook Question

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(x+1)2(x−1)3(x2−10)ƒ(x)=(x+1)^2(x-1)^3(x^2-10)

Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation. x4 + 6x2 + 1 ≥ 4x3 + 4x

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Textbook Question

Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = x2 +2x -8; k=2

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Graph each polynomial function. ƒ(x)=(x-2)2(x+3)

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(4x2+25)/(x2+9)

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=x4+3x3-3x2-11x-6

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