How to Find The Axis of Symmetry

in this video we're going to talk about how to find the axis of symmetry given quadratic equation so first let's talk about two types of parabolas the first one is the equal squared type parabola or variation of that which opens upward and downward so let's say if you have parabola that is in this region here at this point the lowest point on the parabola that is the vertex and let's say it has the coordinates 4 1. now the axis of symmetry is basically line or dashed line rather that cuts the parabola in half now because it's vertical line it's going to be at equals certain number in this case it's the coordinate of the vertex so the axis of symmetry will be is equal to four and in most cases you're dealing with parabolas that open upward or downward but sometimes you might be dealing with parabola that can open to the right or to the left so instead of having an equation that's like equals squared you might get some variation of this type of equation equals squared and it can open to the right or to the left but let's say if we have one that opens to the right the vertex is now at this point and let's say that's at two comma four so the vertex will be two comma four and the line that's going to split the parabola into two equal parts that's going to be horizontal line horizontal dash line instead of vertical dashed line so in this case the axis of symmetry it's going to be the value as opposed to the value when it opens left and right so the axis of symmetry will be basically the equation of this horizontal line and that's equals four so make sure you understand that key difference so when dealing with parabolas that open upward and downward the axis of symmetry is going to be equal the coordinate of the vertex if you're dealing with parabolas that open left and right the axis of symmetry will be the coordinate of the vertex so it's going to be equals whatever number you see here now let's focus on this equation that we have so it's variant of equals squared so we know it's going to open upward or downward so the axis of symmetry is going to be is equal to some number now there's formula that we could use to find the coordinate of the vertex and it's is equal to negative over 2a and this quadratic equation it's in standard form it's in equals ax squared plus bx plus format so is 3 and is negative 12. using that formula is going to be negative times negative 12 over two or two times three negative times negative 12 that's going to be positive 12 2 times 3 is 6 and so 12 divided by 6 is 2. so this is the coordinate of the vertex since we have it already the axis of symmetry is simply is equal to two so that's going to be the vertical line that splits the parabola into two equal parts now let's check our work with this online graphing calculator so i'm going to type in the quadratic equation that we had before equals 3x squared minus 12x plus 7. and we could see the vertex has an coordinate of 2. and the line equals two that is going to split the vertex mean split the parabola into two equal parts and so that is going to be the axis of symmetry as we can see it's vertical since we have parabola that is of the equal squared family now let's try another example so let's say if you're given quadratic equation in vertex form what is the axis of symmetry so we have is equal to two times minus three squared minus five now if you're given it in vertex form you need to be familiar with the standard equation which is equals times minus squared plus the coordinates of the vertex is comma so the first thing recommend you doing is find the coordinates of the vertex basically figure out what and is is this number you see without the negative sign so if you see minus 3 is positive 3 basically change to negative 3 to positive 3. is whatever number you see there you don't have to change the sign so is simply negative five if you set negative three equal to negative will be positive three but if you set negative five equal to positive will be negative 5. so now this is of the equal squared family we know the parabola is going to open upward or downward it's going to open upwards since this is positive which means the axis of symmetry is going to be associated with the coordinate of the vertex so our aos is going to be is equal to 3. now let's check our answer let's delete these two equations and let's write new one is equal to two minus three squared minus five and we said our axis of symmetry is is equal to three and we can see the green line splits the parabola into two equal parts and so that is the axis of symmetry is equal to three now let's try slightly different problem let's say we have this one is equal to three times minus two squared plus 4. go ahead and try that so we have another parabola in vertex form but instead of it being of the equal squared family it's of the is equal to squared family which means it's either going to open to the right or to the left now because we have positive 3 in front and not negative 3 in front of it's going to open to the right now what's the vertex the standard form of this equation is is equal to minus squared plus so the vertex is comma so is simply four if you set equal to positive four that's what it is is four now if you set negative equal to negative two is going to be not negative two but positive two so that's our vertex but because this parabola opens to the right and not up or down the axis of symmetry is going to be associated with the value so our aos is going to be is equal to 2. it's going to be horizontal instead of vertical now let's go ahead and check it so is equal to 3 minus 2 squared plus 4. and we can see it it opens to the right and we said that the axis of symmetry is equals two and that's the line that splits this parabola equally into two parts so that's it for our third example now what if you have quadratic equation in factored form how would you find the axis of symmetry in this case now notice that we have two x-intercepts if you set equal to zero you'll get the x-intercept one and five now it turns out that the vertex or at least the corner of the vertex is going to be the midpoint of these two numbers now if we were to draw graph so we have the intercept at one and at five at equals three we can get the value equals three minus one times three minus five so this is positive two times negative two so that's negative four so the vertex is going to be at three negative four and because we're dealing with equals squared this is going to open upward since it's positive squared so our graph is going to look like that and so our vertex has an value of 3 and value of negative 4. the axis of symmetry is vertical so we're going to focus on the value so our aos is is equal to 3. now let's go ahead and confirm our answer with an online graphing calculator and so we could see the intercept at one and three well put in the wrong one this is supposed to be five so now the intercept is at one and five but the vertex has an x-coordinate of three which is the midpoint of one and five so if you were to add one and five which you get six and then divide by two basically if you average those two numbers you get three and that's how you can find the coordinate of the vertex if you know the two x-intercepts of the parabola now let's go ahead and put in the equation for the axis of symmetry which is equals three and you can see the line it touches the vertex split in the parabola into two equal parts so that's it for this video now you know how to find the axis of symmetry of quadratic equation or even of any parabola