Table of contents

0. Review of Algebra4h 18m

Algebraic Expressions
36m

Exponents
32m

Polynomials Intro
19m

Multiplying Polynomials
36m

Factoring Polynomials
1h 2m

Radical Expressions
15m

Simplifying Radical Expressions
35m

Rationalize Denominator
15m

Rational Exponents
4m

1. Equations & Inequalities3h 18m

Linear Equations
31m

Rational Equations
21m

The Imaginary Unit
6m

Powers of i
11m

Complex Numbers
36m

Intro to Quadratic Equations
18m

The Square Root Property
11m

Completing the Square
12m

The Quadratic Formula
18m

Choosing a Method to Solve Quadratics
9m

Linear Inequalities
20m

2. Graphs of Equations1h 43m

Graphs and Coordinates
7m

Two-Variable Equations
23m

Lines
1h 12m

3. Functions2h 17m

Intro to Functions & Their Graphs
35m

Common Functions
5m

Transformations
45m

Function Operations
21m

Function Composition
29m

4. Polynomial Functions1h 44m

Quadratic Functions
39m

Understanding Polynomial Functions
34m

Graphing Polynomial Functions
30m

Dividing Polynomials

Zeros of Polynomial Functions

5. Rational Functions1h 23m

Introduction to Rational Functions
9m

Asymptotes
35m

Graphing Rational Functions
38m

6. Exponential & Logarithmic Functions2h 28m

Introduction to Exponential Functions
9m

Graphing Exponential Functions
25m

The Number e
8m

Introduction to Logarithms
22m

Graphing Logarithmic Functions
18m

Properties of Logarithms
27m

Solving Exponential and Logarithmic Equations
35m

7. Systems of Equations & Matrices4h 5m

Two Variable Systems of Linear Equations
1h 7m

Introduction to Matrices
1h 11m

Determinants and Cramer's Rule
1h 3m

Graphing Systems of Inequalities
42m

8. Conic Sections2h 23m

Introduction to Conic Sections
5m

Circles
15m

Ellipses: Standard Form
33m

Parabolas
36m

Hyperbolas at the Origin
40m

Hyperbolas NOT at the Origin
12m

9. Sequences, Series, & Induction1h 22m

Sequences
40m

Arithmetic Sequences
25m

Geometric Sequences
15m

10. Combinatorics & Probability1h 45m

Factorials
11m

Combinatorics
46m

Probability
47m

2. Graphs of Equations

Graphs and Coordinates

College Algebra

2. Graphs of Equations

Graphs and Coordinates

Previous problemNext problem

0:50 minutes

Problem 61

Textbook Question

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.

Verified step by step guidance

Step 1: Understand the vertical line test. The vertical line test is a method to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, then the graph does not represent a function.

Step 2: Observe the graph provided. The graph is a red parabola that opens upwards, centered around the y-axis.

Step 3: Apply the vertical line test. Imagine drawing vertical lines at various x-values across the graph. Check if any vertical line intersects the graph at more than one point.

Step 4: Analyze the intersections. For this parabola, each vertical line intersects the graph at exactly one point for every x-value. This indicates that y is uniquely determined for each x.

Step 5: Conclude based on the test. Since no vertical line intersects the graph at more than one point, the graph passes the vertical line test, meaning y is a function of x.

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Key Concepts

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

Vertical Line Test

The vertical line test is a method used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function, as a function can only have one output (y-value) for each input (x-value). This test is essential for identifying functions visually.

Function Definition

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for every x in the domain, there is a unique y in the range. Understanding this definition is crucial for applying the vertical line test effectively.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of mathematical functions. It requires understanding the axes, the shape of the graph, and how points on the graph correspond to ordered pairs (x, y). This skill is vital for applying the vertical line test and determining whether a graph represents a function.