4. Polynomial and Rational Functions

Graph Parabolas

Quadratic Functions

Conic Sections: Parabolas, Part 1

Find a" to determine whether vertex is a maximum or a minimum"

In Exercises 9–16, find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)^2+5

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. f(x) = -x^2 +2x + 3

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. f(x) = (x + 4)^2 - 2

In Exercises 1–4, the graph of a quadratic function is given. Write the function’s equation, selecting from the following options. <p><img src="https://lightcat-files.s3.amazonaws.com/problem_images/f51b9af1b648a60f-1676923843754.jpg" alt="" /></p>

Maximum and Minimum Values of Quadratic Functions

In Exercises 49–52, write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x^2 but with the given point as the vertex. (5, 3)

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. f(x)=4−(x−1)^2

In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-1) The graph passes through the point (-2,-3).

Transformation of y = x^2 and Its Attributes

Find maxima and minima using vertex form

Conic Sections: Parabolas, Part 2  (Directrix and Focus)

Understand horizontal scaling of quadratic equations

Find the vertex of a graph and determine whether it is a maximum or minimum

In Exercises 45–48, give the domain and the range of each quadratic function whose graph is described. The vertex is and the parabola opens up.

In Exercises 9–16, find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2x^2−8x+3

In Exercises 9–16, find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−x^2−2x+8

Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is the minimum product?

Identify features of a quadratic equation

In Exercises 39–44, an equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value.
b) Find the minimum or maximum value and determine where it occurs.
c) Identify the function’s domain and its range. f(x)=−4x^2+8x−3

Application of quadratic function transformations

Graphing Quadratic Functions - Example 1

Among all pairs of numbers whose difference is 24, find a pair whose product is as small as possible. What is the minimum product?

How to find the Maximum & Minimum Values of Quadratic Functions

Graph quadratic functions

Characteristics of Parabolas

Find the Vertex of a Parabola by Using a Formula

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. f(x)=2x^2+4x−3

In Exercises 45–48, give the domain and the range of each quadratic function whose graph is described. Maximum = -6 at x = 10

In Exercises 9–16, find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2(x−3)^2+1

 Formula to Find the Vertex of a Parabola

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. f(x)=2x−x^2−2

Graphing Equations by Plotting Points - Example 1

Graph quadratic functions using the vertex and another point

Optimization Problem

 Graphing a Quadratic Function from Standard Form

Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?

In Exercises 39–44, an equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value.
b) Find the minimum or maximum value and determine where it occurs.
c) Identify the function’s domain and its range. f(x)=3x^2−12x−1

In Exercises 53–56, write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 3x^2 or g(x) = -3x^2, but with the given maximum or minimum. Minimum = 0 at x = 11

Graphing a Parabola

Vertex Form of a Quadratic Function

In Exercises 5–6, use the function’s equation, and not its graph, to find (a) the minimum or maximum value and where it occurs. (b) the function's domain and its range. f(x) = -x^2 + 14x - 106

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. f(x)=x^2−2x−3

Find the maximum or minimum value of a quadratic equation using vertex form: real-world application

Identifying Key Features of a Parabola

In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-4) The graph passes through the point (1,4).

In Exercises 39–44, an equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value.
b) Find the minimum or maximum value and determine where it occurs.
c) Identify the function’s domain and its range. f(x)=5x^2−5x

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. f(x)=x^2+3x−10

Graph of y = x^2 and Its Attributes

How to Find the Maximum or Minimum Value of a Quadratic Function Easily

Find axis of symmetry and vertex

In Exercises 1–4, the graph of a quadratic function is given. Write the function’s equation, selecting from the following options. <p><img src="https://lightcat-files.s3.amazonaws.com/problem_images/5cb19936a8ae1c9b-1677067668469.jpg" alt="" /></p>

In Exercises 53–56, write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 3x^2 or g(x) = -3x^2, but with the given maximum or minimum. Maximum = 4 at x = -2

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. f(x)=x^2+6x+3

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. f(x)=2x−x^2+3

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. f(x) = - (x + 1)^2 + 4

Vertically translate quadratic equations

In Exercises 49–52, write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x^2 but with the given point as the vertex. (−10, −5)

Completing the Square and Vertex Form of Quadratic Equations

Learn the toughest concepts covered in Algebra with step-by-step video tutorials and practice problems by world-class tutors

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1. Fundamental Concepts of Algebra

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Use Long Division to Divide Polynomials

Use Synthetic Division to Divide Polynomials

Evaluate a Polynomial Using the Remainder Theorem

Use the Factor Theorem to Solve a Polynomial Equation

Dividing Polynomials

Identify Polynomial Functions

Recognize Characteristics of Graphs of Polynomial Functions

Determine End Behavior

Use Factoring to Find Zeros of Polynomial Functions

Identify Zeros and Their Multiplicities

Use the Intermediate Value Theorem

Understand the Relationship Between Degree and Turning Points

Graph Polynomial Functions

Polynomial Functions and Their Graphs

Solve Polynomial Inequalities

Solve Rational Inequalities

 Problems Modeled by Polynomial or Rational Inequalities

Polynomial and Rational Inequalities

Find the Domains of Rational Functions

Use Arrow Notation

Identify Vertical Asymptotes

Identify Horizontal Asymptotes

Use Transformations to Graph Rational Functions

Graph Rational Functions

Identify Slant Asymptotes

Solve Applied Problems Involving Rational Functions

Rational Functions and Their Graphs

Recognize Characteristics of Parabolas

Determine a Quadratic Function's Minimum or Maximum

Problems Involving a Quadratic Function's Minimum or Maximum

Solve Direct Variation Problems

Solve Inverse Variation Problems

Solve Combined Variation Problems

Solve Problems Involving Joint Variation

Modeling Using Variation

Use Rational Zero Theorem to Find Possible Rational Zeros

Find Zeros of a Polynomial Function

Solve Polynomial Equations

Use the Linear Factorization Theorem to Find Polynomials With Given Zeros

Use Descartes's Rule of Signs

Zeros of Polynomial Functions

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