Finding the Intercepts of a Graph (by Example)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

(D4M) — Here is the video transcript for this video.

00:00
in this episode you’ll learn what the x and y intercepts are and how to find
them [Music] so first what are intercepts so to find
the x-intercepts of a graph we’re going to let y be 0 and we’re going to solve
the equation for x so let’s go over here and look at an illustration
so let’s look at something that goes up and down like this
and then these right here will be the x-intercepts
we’ll call this the x-axis and the y-axis
so for example what if we have something like y squared equals x to the third
plus 2x plus one and so to find the x intercepts we’re going to let y be 0 here
and then we would try to go solve this and so we would find the x intercepts

00:01
right here so this could be uh you know a point right here x one zero
and this could be another point here x two zero
and this could be the point right here x three zero
and so we would have three x-intercepts so that’s how we find them and and
here’s what there are through an illustration
these are the x-intercepts right here number two to find the y-intercepts
we’re gonna let x be zero and we’re gonna solve the equation for y
so on number 2 so we would we could have something that looks like this
and this right here would be the y-intercept
so it would be the point something like 0 y1 and we could have something like y
equals x to the third plus two x plus one and to find the y intercept here we
would set uh x to be zero so we would say y equals

00:02
zero to the third plus two times zero plus one and then we would go and solve
for for this y here y would be one so this point here would be
1 for the x intercept here 0 1 for the y intercept and so
here’s the here’s another example what if we have this equation here y squared
equals 6 minus x so let’s look at that graph up here
let’s say let’s look at this example and to
we can look at a sketch of this graph right here so it’s going to come through
here like this and it’s going to pass through right here at 6.
so this will be the graph of y squared equals 6 minus x
so what are the x and y intercepts so here how do we get this intercept right
here so or how do we get these two intercepts here so to find the x intercepts

00:03
we can let y be zero so we’ll let y be zero right here so we’re going to get
zero equals six minus x and so if we move the x over we get x
equals 6 so that’s how we get this point right here so 6 0 is the
is the x intercept and how do we find the y intercept
so to find the y-intercept we’re going to let the x be 0. so we’ll let the x be
0 here now so now we’re going to get y squared equals 6 minus 0
or in other words y squared equals 6 and then if i take the positive roots on
both sides right here then i’ll get plus or minus square root of six
and so we can see that we’re going to get these two y-intercepts right here
so we’ll get uh zero square root of six and 0 minus square root of 6 here
and these are the y-intercepts all right so very good so that’s what

00:04
intercepts are x and y-intercepts okay so let’s look at the next thing here and
let’s erase this here real quick so [Music] let’s look at another example here
x to the third minus 4x so here we’re going to look for the x
intercepts first so we’re going to let y be zero so i’m going to go up here and
say 0 equals x to the third minus 4x and so i’m going to factor that here
so let’s say we factor out an x and get x squared minus 4
or x and then x minus 2 and then x plus two so we’re going to get x equals zero
and then x equals two connect x equals minus two
and so now we’re gonna have three uh x intercepts
so we’re gonna have for example we’re going to have 0 0

00:05
then we’re going to have 2 0 then we’re going to have minus 2 0. so
these are the x-intercepts so remember x-intercepts are points so
not equations right so these are the x-intercepts here
now how do we find the y-intercepts so to find the y-intercepts i’m going to
let x be 0 now so i’m going to go this equation right here and let x be zero
and so what do we get here we’re going to get zero to the third
minus four times zero so that’s just y equals zero
so what is the y intercept so the y intercept is when x is zero y zero and
so this is the y intercept okay so there’s another example there finding
the intercepts and what they are and so let’s look at the next example
now so let’s do one with absolute value here let’s erase this real quick

00:06
all right so let’s look at absolute value now
and so again i’m going to be looking for x-intercepts so i’m going to set the y
equal to 0. so i’m going to get up i’m going to get
this equation here i’m going to get 0 for the y and then equals absolute
value of 3x minus 7. so that gives me 0 equals 3x minus 7.
and then if i move the 7 over and then divide by 3 and so i get this for my x
and so my x intercept is going to be 7 3 0 that’s my x intercept
and what about the y-intercept so to find the y-intercept i need to let x be
zero so here we go y equals ah absolute value of three times the x is zero
and so if we look at that that’s just absolute value minus seven which is seven
so here we’re going to have zero seven is the y-intercept

00:07
okay so there’s um another example there finding the y x and
y-intercepts now let’s ask some questions here
so the question is can a graph have can a graph have
no intercept so that’s the first question can a graph have no no
intercepts at all so i’ll draw a picture right here to convince you
so let’s look at something like how about like that so this could be
something like square root of x minus two plus
so it looks like square root function but it’s been shifted around a little
bit but anyways it’s going to start right here and just go up
and so then it would have no intercepts right there what about can a graph have
many x intercepts and no y-intercepts so let’s draw something that looks like
that so we can so it has to have no y-intercept
so i’ll put a hole right here through the y-axis

00:08
and i’ll start it here and then i’ll just make it go
and it has many x-intercepts so i’ll make it go go up and down [Music]
many times so yeah the answer is yes and i think for number two is also
yes so number two you can look at something like cosine x
if you know what trig functions are then you can just say x is greater than zero
so that you know just draw something that looks like cosine cosine goes up
and down the highest point is one and then minus one and it’ll have
infinitely many x-intercepts so yeah it could have many many
what about no x-intercepts and only one y-intercept
um so then we can do something like number three here
we can do something like how about y equals x squared plus one
so that’s shifting the x squared but up one looks like this

00:09
so it has one y-intercept and no x-intercepts here and what about number four
many x intercepts and only one y intercept that one’s easy we’ll just
adapt the second one we just did except this time we won’t put a hole in it
we’ll just go it goes up and down over and over again this is y equals cosine x
it only has one y intercept and it has many x intercepts
and what about number five can a graph have many x intercepts
and two y-intercepts and so for something like that um
you could say something like this now because we’re just asking can a
graph we’re not asking can a function in the graph of a function we’re just
having the graph of anything right so many x-intercepts and two y-intercepts

00:10
so we could take something like the cosine graph goes up and down up and down um
and then we can put um many x-intercepts um but it doesn’t have two y-intercepts
so we’ll just add another point to the graph so that would have two y intercepts
and then so it would look like cosine x like that function right there but then
we’ll throw in another function but then another point but then it’s not
a function but the problem doesn’t ask specifically for a function right here
so yeah you could have two y intercepts as long as you’re not trying to have a
function all right so now the next question is
what about if you’re asking specifically
about a function what are the cases that can happen
so let’s look at that real quick now we’re only looking at graphs of functions

00:11
and we’re going to ask the question can we have you know no intercepts
so number one and actually the number one that we gave back on the last slide
um satisfied this condition right here so this was a function right here this
is the graph of a function and it has no intercept so it’s just the
square root function that’s been shifted to the right two
and it’s been shifted up three it has no intercepts this is the graph
of a function alright so what about many x-intercepts and no y-intercept
and so again we can do something like this time we’ll say something how about
like a sign but i’m going to say no y intercept so
i’m going to take away the origin and so it’s just going to go up and down
so this will be 1 and minus 1. and so we’ll say y equals sine x

00:12
um but we’ll make it a piecewise function and we’ll take away zero
so when x is not zero all right and so no y-intercept so
this is our function here sine x as long as x is not zero here
so that would give infinitely that would give many x-intercepts because it just
keeps going the same pattern over and over again up and down up and down and
yeah so there’s number two and so number three
um what about number three so number three is no x-intercepts and
uh and only one y-intercept so can we do a function for that
and so we can look at something like um x squared plus one
so this is a function right here has no x-intercepts and it only has one
y-intercept and so that’s just the parabola y equals x square shifted up one

00:13
and what about number four number four says can it have many x-intercepts and
only one y-intercept and so that one we don’t need to do the
restriction we can just look at a sine function it just goes up and down up and
down over and over again and it only has one y intercept right there
so this will just be the regular sine function um and then number five
and so can it have two y-intercepts and so no
and the answer is no because any two anytime you have two y intercepts it’s
not going to be a function so no not a function [Music] whenever there are
two y intercepts okay when there are two y-intercepts okay so um let’s look at a

00:14
and another example now so let’s look at this function right here
x to the third minus four x squared let’s look at that real quick so
to find the x-intercepts we’re gonna let y be zero
so let’s go up here and say okay y is what 2x to the third minus 4x squared
and we’re going to let y be zero so i’m going to factor out of here a 2
and an x squared and i have an x left and another 2 left here
so it looks like we’re going to get x equals 0 and x equals 2
and so we’re going to get some intercepts here 0 0 and 2 0.
so these are the x-intercepts and what about y-intercepts so to find
the y-intercepts we’re going to let x be 0. so here we go we’re going to say y

00:15
equals 2 times 0 to the third minus 4 times 0 squared
and when we calculate all that we get 0. so 0 0 is the y-intercept
in fact now that you know that whenever you’re looking at a function there’s
only one y-intercept as soon as you know this is the x-intercept that’s also a
y-intercept so really this right here you already kind of knew it just by
looking at that and the fact that that’s a function right there but just in case
you didn’t recognize that there’s always a solid way of going to find the y
intercepts just let x be zero and then you can find the same same thing there
okay and so now the last thing i want to point out is um what if we have um
something that looks harder so let’s look at this real quick

00:16
what if it’s not possible to find um the x and y intercepts there
so let’s look at two x to the fifth minus four x squared plus three
all right let’s just go about the business of trying to find the
x-intercepts and y-intercepts so to find the
x-intercepts we’re going to let y be 0 just like we did before so y
which turns out to be the 2x squared minus 4x squared plus 3
and i’m gonna let y be zero on the last example it happened to
factor nicely and it worked out but actually how do you solve this right here
so this could be difficult to solve so that’s why i’m saying if possible
it may not always be possible to find your x-intercepts
and in this case right here it can be difficult to find those x-intercepts
you may need to resort to approximating if you were to use
say some software or some advanced mathematics like calculus then you could

00:17
get an approximation or if you use a calculator or some
some kind of technology you would get an approximation um
you would get that and then you would get four other imaginary or complex roots
that have non-zero imaginary parts but anyways the x-intercepts can be hard
to find so that’s what i want to point out here at the end of this video is the
x-intercepts can be hard to find can be hard to find
it’s not always guaranteed that you can just solve any equation and find the
exact value like that well what about the y intercepts the y
intercepts on the other hand is when you plug in x to zero
and that is the opposite that’s really just a plug in to your function
and you know in fact not only is it doable but it’s
just actually very easy zero zero zero that’s just three

00:18
so zero three is the y-intercept and there we go
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About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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