MCR3U (1.1) - Relations vs Functions - Grade 11 Functions

Summary

In this educational video, Patrick explores the concept of functions by examining various mathematical relations to determine if they qualify as functions. He explains the vertical line test, a method to verify if a relation is a function by ensuring no vertical line intersects the graph at more than one point. Patrick covers linear equations, circles, parabolas, and a sideways parabola, demonstrating through examples and graphical illustrations how each relation either passes or fails the test. The video is a comprehensive guide for understanding the properties of functions in algebra.

Takeaways

• 📐 The video discusses how to determine if different relations are functions or not.
• ✅ The relation y = 3 - x is a function because it passes the vertical line test, indicating a unique y for each x.
• ⛔ A vertical line, such as x = 4, is not a function as it fails the vertical line test, having multiple y values for the same x.
• 🔵 The equation x^2 + y^2 = 16 represents a circle and is not a function because it does not pass the vertical line test, having two y values for each x.
• 📉 The parabola y = x^2 - x - 6 is a function as it passes the vertical line test, with a unique y for each x, and opens upwards due to the positive leading coefficient.
• 🔄 The relation x = y^2 - 1, when rearranged to y = ±√(x + 1), is not a function because it fails the vertical line test, having two y values for each x greater than or equal to -1.
• 📊 The video emphasizes the importance of graphing relations to visually confirm whether they are functions by applying the vertical line test.
• 📋 The script reviews the general forms of lines, circles, and parabolas, and how their properties relate to the vertical line test.
• 🚫 A sideways parabola, like the one in the fourth example, is not a function because it has multiple y values for the same x.
• 📘 The video suggests that if unsure, one should review how to graph different types of functions from previous math courses to better understand and apply the vertical line test.

Q & A

What is the relation described by the equation y = 3 - x?

The relation described by the equation y = 3 - x is a linear function. It represents a straight line with a slope of -1 and a y-intercept of 3.

How can you determine if the line y = 3 - x is a function?

You can determine if y = 3 - x is a function by applying the vertical line test. Since no vertical line intersects the graph of this equation more than once, it passes the test and is indeed a function.

Why is a vertical line not considered a function?

A vertical line is not considered a function because it fails the vertical line test. There are multiple points on the line that can be intersected by a single vertical line, indicating that there are multiple y-values for a single x-value.

What type of geometric shape does the equation x^2 + y^2 = 16 represent?

The equation x^2 + y^2 = 16 represents a circle with a radius of 4 units, centered at the origin of a Cartesian coordinate system.

Why does the circle defined by x^2 + y^2 = 16 fail the vertical line test?

The circle defined by x^2 + y^2 = 16 fails the vertical line test because for any given x-value, there are two corresponding y-values (one positive and one negative), indicating that it is not a function.

What is the relation represented by the equation y = x^2 - x - 6?

The relation represented by the equation y = x^2 - x - 6 is a quadratic function, which graphs as a parabola. It factors into (x - 3)(x + 2), indicating x-intercepts at x = 3 and x = -2.

Is the parabola y = x^2 - x - 6 a function? How do you know?

Yes, the parabola y = x^2 - x - 6 is a function. It passes the vertical line test because for any given x-value, there is only one corresponding y-value.

What is the equation x = y^2 - 1 transformed into when solved for y?

When the equation x = y^2 - 1 is solved for y, it becomes y = ±√(x + 1). This transformation indicates that y is defined for all x values greater than or equal to -1.

Why does the graph of x = y^2 - 1 fail the vertical line test?

The graph of x = y^2 - 1 fails the vertical line test because for certain x-values, there are two possible y-values (one positive and one negative), which means there are multiple y-values for a single x-value.

What is the key difference between a function and a non-function in terms of the vertical line test?

The key difference is that a function will pass the vertical line test, meaning that no vertical line will intersect the graph of the function more than once. A non-function will fail this test, as there will be at least one vertical line that intersects the graph at more than one point.

What general rule can be applied to determine if a line is a function?

The general rule is that any line that is not vertical is a function because it will pass the vertical line test. A vertical line, on the other hand, is not a function as it fails the test.

Outlines

00:00

📚 Understanding Functions and Graphs

This paragraph introduces the concept of determining whether various mathematical relations are functions or not. Patrick walks through the process of identifying if a given relation is a function by using the vertical line test. He starts with a linear equation, y = 3 - x, and explains that it represents a line with a slope of -1 and a y-intercept of 3. Patrick then demonstrates that this line passes the vertical line test, confirming it is a function. He emphasizes that every line, except for a vertical line, is a function. He also touches on the fact that a circle, represented by the equation x^2 + y^2 = 16, does not pass the vertical line test due to multiple y-values for a single x-value, thus it is not a function.

05:02

📘 Analyzing Parabolas and Special Cases

In the second paragraph, Patrick continues the discussion on functions by examining a parabola defined by the equation y = x^2 - x - 6. He factors the equation to show the x-intercepts and explains that the parabola opens upwards due to the positive leading coefficient. Patrick then asserts that this parabola passes the vertical line test, making it a function. He contrasts this with a sideways parabola, represented by the equation x = y^2 - 1, which he re-arranges to y = ±√(x + 1). Patrick creates a table of values to illustrate that this sideways parabola fails the vertical line test at multiple points, indicating that it is not a function. He concludes by suggesting that practicing graphing and applying the vertical line test is key to determining the function status of various mathematical relations.

Keywords

💡Function

A function in mathematics is a relation where each input (or independent variable) is associated with exactly one output (or dependent variable). This is a central concept in the video, as the host is determining whether various relations are functions or not. The video uses the vertical line test to verify if a relation is a function; if any vertical line intersects the graph of the relation at more than one point, it is not a function. Examples from the script include the line y = -x + 3, which passes the vertical line test, confirming it is a function.

💡Vertical Line Test

The vertical line test is a graphical method used to determine if a relation is a function. If a vertical line drawn anywhere on the graph intersects the graph at more than one point, the relation is not a function. This test is repeatedly applied throughout the video to various relations, such as lines, circles, and parabolas, to ascertain their functional status.

💡Line

In the context of the video, a line refers to a linear equation, typically in the form y = mx + b, where m is the slope and b is the y-intercept. The video explains that every line, except for a vertical line, is a function because it passes the vertical line test. An example given is y = -x + 3, which is a downward-sloping line and is confirmed to be a function.

💡Circle

A circle is represented by the equation x^2 + y^2 = r^2, where r is the radius of the circle. The video explains that a circle is never a function because it fails the vertical line test; for any given x-value, there are two corresponding y-values, which violates the definition of a function. The script uses the equation x^2 + y^2 = 16 as an example of a circle.

💡Parabola

A parabola is a type of curve that is symmetrical and has a single point of symmetry called the vertex. The video discusses that parabolas are generally functions because they pass the vertical line test. The script provides an example of a parabola with the equation y = x^2 - x - 6, which factors into (x - 3)(x + 2), indicating its x-intercepts and the direction it opens.

💡Slope

Slope in a linear equation represents the steepness of the line and is the coefficient of x in the equation y = mx + b. The video mentions that a negative slope, as in the line y = -x + 3, means the line slopes downward. The concept is used to describe the direction and steepness of lines in the video.

💡Y-intercept

The y-intercept is the point where a line or function crosses the y-axis. It is represented by the value b in the equation y = mx + b. The video uses the y-intercept to describe the position of the line on the graph, such as in the line y = -x + 3, where the y-intercept is 3.

💡X-intercept

The x-intercept is the point where a graph of a function or relation crosses the x-axis. The video discusses finding x-intercepts by setting y to zero and solving for x, as seen with the parabola y = x^2 - x - 6, which has x-intercepts at x = 3 and x = -2.

💡Quadratic

A quadratic equation is a second-degree polynomial equation of the form ax^2 + bx + c = 0. The video briefly touches on quadratics, mentioning that they will be covered in more detail later. The parabola y = x^2 - x - 6 is an example of a quadratic function discussed in the script.

💡Sideways Parabola

A sideways parabola is a parabola that opens to the left or right instead of up or down. The video explains that sideways parabolas, like the one derived from the equation x = y^2 - 1, fail the vertical line test and are not functions because they have multiple y-values for a single x-value.

Highlights

Introduction to determining whether relations are functions or not.

Explanation of the relation y = 3 - x as a linear function.

Rewriting the line equation in the form y = mx + b to identify the slope and y-intercept.

Visual representation of the line with a negative slope and a y-intercept of 3.

Using the vertical line test to confirm y = 3 - x is a function.

General rule that every line except a vertical line is a function.

Definition of a circle using the equation x^2 + y^2 = r^2 and its graph.

Determining the radius of a circle from its equation.

Failure of the vertical line test for circles, indicating they are not functions.

Introduction to the parabola y = x^2 - x - 6 and its factors.

Graphical representation and orientation of the parabola based on the value of 'a'.

Passing the vertical line test for parabolas, confirming they are functions.

Transformation of the equation x = y^2 - 1 to isolate y.

Graphical representation of the sideways parabola x = y^2 - 1.

Failure of the vertical line test for the sideways parabola, indicating it is not a function.

Suggestion to graph functions to visually confirm if they pass the vertical line test.

Transcripts

what's up guys welcome back Patrick here

and this video we have to determine

whether each of these relations is a

function or not so let's go through them

one by one so we got number one here y

equals three minus X now what kind of

relation is this notice that this is

just a line so we can rewrite this line

as negative x plus 3

all right so y equals MX plus B form so

the B value is 3 that is what that's

always the y intercept as a review and

then there's like a negative 1 in front

of the X so that there is the slope so

if we were to roughly draw this the

y-intercept would be 3 so that would be

up here and then the slope would be

negative 1 so it's like negative 1 over

1 so you would rise down by 1 and then

run by 1 rise down by one run by 1 so

roughly this line would look something

like that right a downward sloping line

and then this is at 3 if you want

something more precise you can throw

this in a table of values pick a bunch

of x-values plot them etc etc I'm just

drawing a rough diagram for now so this

is the diagram for this line is it a

function or not well let's run the

vertical line test if we run a vertical

line through this notice that there's no

points that are going to touch twice on

this line on the vertical line so that

means that this is a function in fact

you may want to make a note that every

line is a function except for a vertical

line so let's say we had a vertical line

maybe let's say x equals 4 so x equals 4

would look something like this a

vertical line is not a function because

notice there's multiple points on this

line that's touching a vertical line

when we run it through but any other

line other than a vertical line is

always going to be a function whether a

downward sloping line or a positive

sloping line even a horizontal line so

let's say this line would be like y

equals I don't know negative 2 that's

going to be a function because it passes

the vertical line test this positive

line passes the vertical line test the

only type of line that is never a

function is a vertical line because

that's going to fail the vertical line

test ok so that's number one number two

we got x squared plus y squared equals

16 if you remember this is a circle

right so we've got x squared plus y

squared equals R squared that is the

general format for a circle so if we

want the radius of this circle we were

just square root 16 which would give us

4 so if we were to graph this the

intercepts would be 4 4 negative 4

negative 4 and you would just draw a

circle like that right so this graph

here is x squared plus y squared equals

16 so is this a function or not run the

vertical line test no it's not a

function let's notice here and here for

example at that x value there's two y

values at the next x value there's two Y

values right so there's so many times

where there's two Y values for an x

value so we know that this is not a

function and in fact in general a circle

is never a function it's always going to

fail the vertical line test so whenever

you see something in this format you

know right away it's not a function

because it's a circle but if you want to

show visually you can graph it and just

say that it doesn't pass the vertical

line test number three we got y equals x

squared minus X minus 6 if you remember

this is a parabola here and this

actually factors smoothly this factors

into X minus 3 X plus 2 if you remember

from great n we're gonna be doing a

whole unit on quadratic so I'm not going

to go into too much detail right now

about how to graph this but this is

still from grade 10 so from here

this factors into that x-intercepts x

equals three x equals negative two so

negative two over here positive three is

over there and the a value is one so

this parabola is going to open up

because the a values positive so this

roughly is gonna look something like

that

right not exact again if you want you

can make a table of values if you want

an exact graph but it's gonna look

something like this

is it a function or not if we run a

vertical line through it notice that

it's gonna pass the vertical line test

there's not going to be any points any y

values any two y values for one x value

so this is a function it passes the

vertical line test and in fact in

general every parabola is always going

to be a function unless it's a sideways

parabola like number four but we will

get to that but any normal parabola

whether it opens up whether it opens

down it's always going to be a function

so always going to pass the vertical

line test then number four we got x

equals y squared minus one this one's a

little bit weird okay I'm gonna do this

I'm actually gonna isolate for Y here so

I'm gonna bring the negative one over so

I'll have X plus one equals y squared

and then square root both sides to get

the Y by itself and remember the square

root of something it's always plus or

minus so I'm actually gonna make a table

of values for this notice that the

square root under the square root you

can never have a negative number right

you can't square root of negative

numbers so notice that X values they all

have to be negative one or more right

because if it's less than negative one

so for example if we plug in negative 2

for X negative two plus one is negative

one can take the square root of negative

one so all the X values are going to be

at least negative one or greater than

that so if X is negative one

what is Y going to be well negative one

plus one is zero square root of zero is

just zero what if

we plug in zero here for X well 0 plus 1

is 1 square root of 1 is plus or minus 1

so there's going to be two y values that

are plus or minus 1 what if we plug in 1

for X well if we plug in 1 for X we

actually won't get a smooth number for y

because it's gonna be 1 plus 1 which is

2 and the square root of 2 is some kind

of decimal so let's pick X values where

we're gonna continue having smooth

numbers so 3 3 works so if I plug in 3

for X 3 plus 1 is 4 square root of 4 is

plus or minus 2 and that's actually

enough points to create a graph for now

so if I draw this out negative 1 and 0

that is here 0 and plus or minus 1 so 0

plus 1 0 minus 1 that's here and then 3

plus or minus 2 that's like here and

here right that's 3 and this is negative

2 this is positive 2 so again this is

just a sideways parabola when you graph

it ok and then is this a function or not

run the vertical line through it notice

it's going to fail everywhere basically

there's multiple dependent variables for

independent variables so 0 the

independent variables 0 has two

dependent variables plus 1 and minus 1/3

that independent variable has two

dependent variables plus 2 minus 2 so

it's going to be failing the vertical

line tests so it is not a function so if

you get questions like this my

suggestion is to try and graph them in

different ways you may have to go back

to grade 10 a bit review how to graph

certain functions but once you graph

them you can run the vertical line test

then you could see is it a function or

not