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