Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 49

Chapter 4, Problem 49

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 guidance

Recall the quadratic function is given by $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Understand the given conditions: $a > 0$ means the parabola opens upward, so the graph should be a 'U' shape opening upwards.

Use the discriminant formula $\Delta = b^2 - 4ac$ to determine the nature of the roots. Since $b^2 - 4ac > 0$, the quadratic has two distinct real roots.

Interpret these conditions graphically: the parabola opens upward and crosses the x-axis at two distinct points (because of two real roots).

From the given graphs A–F, select the one that shows a parabola opening upward and intersecting the x-axis at two distinct points.

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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.

Discriminant of a Quadratic Equation

The discriminant, given by b² - 4ac, indicates the nature of the roots of a quadratic equation. If the discriminant is greater than zero, the equation has two distinct real roots, meaning the parabola intersects the x-axis at two points. This helps in identifying the correct graph based on root behavior.

Effect of Coefficient 'a' on Parabola Orientation

The coefficient 'a' in the quadratic function determines the direction the parabola opens. When a > 0, the parabola opens upward, resembling a 'U' shape. This is crucial for selecting the correct graph since the problem specifies a > 0, so the parabola must open upwards.

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Related Practice

Textbook Question

For each polynomial function, find all zeros and their multiplicities. $ƒ (x) = 3 x (x - 2) (x + 3) (x^{2} - 1)$

Textbook Question

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=5x²(x²-16)(x+5)

Textbook Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=3x²-x-4; 1 and 2

Textbook Question

Graph each polynomial function. ƒ(x)=2x³+x²-x

Textbook Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=-2x³+5x²+5x-7; 0 and 1

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) = x³ - 3x² + 4x -4; k=2

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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 video answer for a similar problem:

Key Concepts

Quadratic Function and Its Graph

Discriminant of a Quadratic Equation

Effect of Coefficient 'a' on Parabola Orientation

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