Subjects

Algebra 2

Transformations

Shifting Functions

Graphing Shifted Functions

•

Dec 1, 2025

•

2 pages

Understanding Graphs: Relations and Functions Simplified

Chelsea

@zuncho

Functions and relations are key concepts in algebra that describe... Show more

Relations and Functions

A relation is simply a set of ordered pairs that connect input values to output values. A function is a special type of relation where each input $x-value$ has exactly one output $y-value$ . Think of a function as a machine that takes one number in and produces exactly one number out.

When working with functions, we need to know the domain (all possible input values) and the range (all possible output values). These tell us what numbers can go into our function and what numbers can come out.

Let's look at examples like $y^2 = x$ . When we graph this relation, we see a sideways parabola. Since some x-values connect to two different y-values, this is not a function. Its domain is $x \geq 0$ and its range is all real numbers.

💡 A quick way to remember: In a function, you can never have two different y-values for the same x-value. It's like a rule that says "no x-value can have multiple partners."

Similarly, when we graph $|y| = x$ , we get two lines forming a V-shape. Again, some x-values have two matching y-values, so this is a relation but not a function. Its domain is $x \geq 0$ and its range is all real numbers.

The Vertical Line Test

The vertical line test is a super helpful trick to determine if a graph represents a function. If any vertical line crosses the graph more than once, it's not a function. This works because multiple intersections mean one x-value has multiple y-values.

When analyzing relations presented as ordered pairs, check if any x-value appears more than once. If it does and the y-values are different, it's not a function. For example, if (-1, 4) and (-1, 1) both appear in a relation, it's not a function because the input -1 has two different outputs.

When examining graphs, pay close attention to their shapes. Circles, sideways parabolas, and horizontal lines typically fail the vertical line test. Vertical parabolas, sloped lines, and curves that don't loop back typically pass the test and represent functions.

🔍 When identifying domains and ranges from graphs, look for the leftmost and rightmost points to find the domain limits, and the highest and lowest points to find the range limits.

Remember that domain and range can be written using inequality notation $like $x \geq -7$$ or interval notation. Being able to quickly identify whether a relation is a function and determine its domain and range will help you solve more complex problems in algebra and calculus.

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Subjects

Algebra 2

Transformations

Shifting Functions

Graphing Shifted Functions

Algebra 2

•

Dec 1, 2025

•

2 pages

Understanding Graphs: Relations and Functions Simplified

Chelsea

@zuncho

Functions and relations are key concepts in algebra that describe how values relate to each other. Understanding the difference between relations and functions will help you make sense of mathematical relationships and prepare you for more advanced math topics.

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Relations and Functions

💡 A quick way to remember: In a function, you can never have two different y-values for the same x-value. It's like a rule that says "no x-value can have multiple partners."

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The Vertical Line Test

🔍 When identifying domains and ranges from graphs, look for the leftmost and rightmost points to find the domain limits, and the highest and lowest points to find the range limits.