Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 45

Chapter 4, Problem 45

Give the domain and the range of each quadratic function whose graph is described. The vertex is and the parabola opens up.

Verified step by step guidance

Identify the vertex of the quadratic function, which is given as a point \( (h, k) \). This point represents the minimum point of the parabola since it opens upward.

Recall that the general form of a quadratic function with vertex \( (h, k) \) is \( f(x) = a(x - h)^2 + k \), where \( a > 0 \) because the parabola opens up.

Determine the domain of the function. Since quadratic functions are defined for all real numbers, the domain is \( (-\infty, \infty) \).

Determine the range of the function. Because the parabola opens upward and the vertex is the minimum point, the range is all \( y \)-values greater than or equal to \( k \), so the range is \( [k, \infty) \).

Summarize: Domain is \( (-\infty, \infty) \) and range is \( [k, \infty) \), where \( k \) is the \( y \)-coordinate of the vertex.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Quadratic Functions and Their Graphs

A quadratic function is a polynomial of degree two, typically written as f(x) = ax² + bx + c. Its graph is a parabola, which can open upward or downward depending on the sign of the coefficient a. Understanding the shape and orientation of the parabola is essential for analyzing its properties.

Vertex of a Parabola

The vertex is the highest or lowest point on the parabola, representing either a maximum or minimum value of the quadratic function. It is given by the coordinates (h, k) in vertex form f(x) = a(x - h)² + k. Knowing the vertex helps determine the function’s range and the axis of symmetry.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since you can input any x-value. The range depends on the vertex and the direction the parabola opens: if it opens upward, the range is all y-values greater than or equal to the vertex’s y-coordinate; if downward, all y-values less than or equal to that coordinate.

Recommended video:

4:22

Domain & Range of Transformed Functions

Related Practice

Textbook Question

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. g(x)=1/(x−1)

Textbook Question

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=3x⁴−11x³−x²+19x+6

Textbook Question

In Exercises 45–46, describe in words the variation shown by the given equation. z = kx^2 √y

views

Textbook Question

Give the domain and the range of each quadratic function whose graph is described. Maximum = -6 at x = 10

Textbook Question

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)/(x+4)<0

views

Textbook Question

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. x⁴−3x³−20x²−24x−8=0

views