When graphing equations, intercept and
symmetry help give us important information
to graph that equation.
Let's look at this example
where we have the graph.
So the graph is given, and we want to
just list the intercepts.
Now the intercepts are points on the graph
that intersect either the x or the y-axis.
So let's look at this graph.
Does the graph, which is here and here,
ever intersect the x-axis?
Well, you can see no, so of course
there are no x-intercepts.
We'll write none.
Now we'll look for the y-intercepts.
We're going to look for the points where the
graph intersects the y-axis.
Well we can see two points,
so we need to list them.
We need to be careful that we look at the
tick marks along the axis: it's not 1, 2, 3;
in this case, it's 2, 4, 6.
So the y-intercepts are
both 6 and negative 6.
Now we're going to look at symmetry.
If a graph has symmetry, it helps give us
extra points on the graph.
So the first thing we want to know is is
this graph symmetric with respect
to the x-axis.
Well let's take a look.
If it's symmetric to the x-axis, for every
point (x , y) on the graph then the point x,
the opposite of y would
also be on the graph.
You can see also that if I were to take this
graph and fold it along the x-axis, then
it's going to match up. So this graph does
have symmetry with respect to the x-axis.
Now let's see if it's symmetric
with respect to the y-axis.
Well let's take a look: if we were to fold
along the y-axis, this particular graph is
going to meet up again. This also means
that for every point (x, y) on the graph, then
the opposite x, the same y is also on the graph.
So this graph has symmetry with respect to
both the x and the y-axis.
Last, let's look and see if this graph is
symmetric with respect to the origin.
And what that means is if I were to say
reflect this graph about the x and then the
y, the graph is going to meet up. Or
vice-versa, reflect on the y and then the x.
You can also see that if I draw a line
through the origin from any point (x, y) on
the graph, then another point that's going
to be on the graph is the
opposite x and the opposite y.
So you can actually see this line, and so
the distance from the origin out to this
point would be the same as the distance from
the origin out to this other point.
So this particular graph is symmetric with
respect to both the x,
the y, and the origin.
So we'll write that down.
Okay, so in this particular example, we were
given the graph, and we found the intercepts
and symmetry from the graph.
Next, we're going to look at finding
intercepts and symmetry from the equation.
Now let's look at finding intercepts and
symmetry given the equation.
Let's look at this equation here: x plus y
squared minus 16 equals 0.
We want to find the intercepts.
Remember those are the points where the
graph intersects the axes.
To find the x-intercepts, if there are any,
what you're going to do is you’re going to
set y equal to 0 and solve for x because, of
course, if you're a point on the x-axis,
your y-coordinate is 0.
So we're going to set y equal to 0, and, of
course, in this equation, we're going to get
x minus 16 equals 0.
So x equals 16 is my x-intercept.
Now we're going to find the y-intercept, and
we're going to set the x equal to zero and
solve for y and see if
there are any y-intercepts.
Okay, in this equation, if we set x to 0,
we're going to get y squared
minus 16 equals 0.
I'm going to add 16 to both sides, take the
square root, and I get plus or minus 4 for
my two y-intercepts.
Now we're going to test for symmetry to see
if it's symmetric with respect to the x, the
y, or the origin.
So we're going to do x-axis, and here we're
going to replace y
with a negative y in the equation.
And what we're looking for is to see if we
get the same equation.
So let's see: if we replace y with negative
y, be careful that you make sure that you
plug in the negative y and use parentheses.
So what happens when I square the negative y
is I still get y squared.
So I end up with x plus y squared
minus 16 equals 0.
Now if we get the same equation, then we
know that this particular graph is symmetric
with respect to the x-axis.
So here, x-axis, we’re going to say yes.
Now we're going to test to see if it's
symmetric with respect to the y-axis.
Now here we're going to replace the x with a
negative x. We're going to get
negative x plus y squared minus 16 equals 0.
Well we can see that in this particular
example, we do not get the same equation, so
what that tells me is that this particular
graph is not symmetric
with respect to the y-axis. So y-axis, we'll write no.
Last, we'll check to see if the graph is
symmetric with respect to the origin.
Here we're going to replace x with negative
x and y with negative y, so again, we'll
write negative x plus negative y squared
minus 16 equals 0.
Again, when we square the negative y, we get
y squared. We end up with negative x plus
y squared minus 16 equals 0.
Again, we compare this to the original
equation; we see that we don't get the same
equation, so this particular equation is not
symmetric with respect to the origin.
So origin, we'll say no.
Okay, let's look at another example just
like this where we have the equation and,
we'll find intercepts and symmetry.
Let's look at this example where we have the
equation and we want to find the intercepts
and test for symmetry.
Well again, to find the x-intercepts, if
there are any, we're going to set y equal to
zero and solve this equation.
And so we get 9x over
x squared plus 25 equals 0.
Well, to solve that equation, I'm going to
multiply both sides by x squared plus 25.
We get 0 equals 9x, so x equals 0.
That is my x-intercept.
Notice that's the origin because
when y is 0, x is 0.
Now we're going to find to see if
there're any y-intercepts.
Here, we're going to set x to 0. So what we
get is 0 over 25, which of course is
0, which we should expect because again
since we knew the origin was an x-intercept.
That is also the y-intercept as well, so the
only point we have for the intercepts is the origin.
Now we will test for symmetry to see if this graph is symmetric with respect to the x, 000:08: 11.550 --> 000:08:15.150 the y, or the origin.
What we're going to do to test for symmetry
with respect to the x is we're going to
replace y with negative y and see if we get
the same equation.
So here we go: we're going to
replace y with negative y.
Now look at this equation, look at the
original: they are not the same equation, so
this particular equation is not
symmetric to the x-axis.
Now we'll test to see if the graph is
symmetric with respect to the y-axis.
So here, we're going to replace x with a
negative x, and let's see what we get.
So we get negative 9x in the numerator, and
we get x squared plus 25 in the denominator.
When we look at the original equation, we
can see that it's not the same equation, so
this one is not symmetric
with respect to the y-axis.
Finally, we're going to check to see if the
graph is symmetric
with respect to the origin.
Here, we replace x with a negative x and y
with a negative y.
We get negative y on the left. We get
negative 9x in the numerator. Again,
we replace this with a negative x.
We're still going to get x squared plus 25.
Now, you would think that the answer is no
because right now, it's not the same
equation. But notice that there's a negative
on the left, and a negative
on the right. So I can divide both sides by
negative 1, and get y equals 9x over
x squared plus 25.
And you can see that, of course, this is the
same equation as the original, so this
particular equation is symmetric with
respect to the origin.
So, origin is yes.
Okay, when graphing equations, intercept and
symmetry help us get important information,
so you need to practice finding intercepts and symmetry.