Over here and the graph is going to be the same thing Gonna be the exact solutions that we just saw right These are actually theĮxact same function. The exact same thing as what we have in blue right over here. Four plus two is six andįour times two is eight. I encourage you to watch videos on factoring polynomials, What adds up to six and when you take their product is eight? Well, four and two. And so, this can be written as, and if you have trouble doing this, Plus six x, plus eight is equal to zero, it willīe useful to factor this. Hopefully, we're getting the hang of this. And so we can sketch out what the graph of y is equal to f of x Make sure we see that, so this is negative one right over So, we have zeros there, negative two, be careful. And we see that we have zerosĪt x equals negative two and x is equal to negative four. Happening for negative x's, so I'll draw it a littleīit more skewed this way. Going to be the vertical line x is equal to negative three. Three, f of x is going to be, let's see, it's going to be negative one times one, right? Negative three plus two is Two plus negative four, over two, so that wouldīe negative six, over two, which is just negative three. X coordinate of the vertex, is going to be halfway in between these. Subtract two from both sides, when x is negative two,Īnd if we subtract four from both sides, or when x If x plus two is equal to zero or x plus four is equal to zero. To zero, if we say x plus two times x plus four is equal to zero, well that's going to happen So, what are the zeros? Well, if you set this equal That's really just the tĬoordinate of the vertex. So, the equation of that line of symmetry is going to be t is equal to five. Well, the line of symmetry is going to be the vertical line that something like that That's the graph of y is equal to f of t. Going to look something like let me draw it a little We can graph f of t, or we can graph y is equal to f of t. Let me make that a littleīit in, t equals two. And then, we know we have zeros at t equals eight and t equals two. So, this is, t is equal to five and y is equal to negative nine, so that's the vertex right over there. Well, we know the vertex is at the point five comma negative nine. And that is my, let's call that my y axis. So, that is our t axis, not our x axis, I have to keep reminding myself. Now, we actually have a lot of information if we wanted to draw it. So, just like that, we haveĮstablished the vertex. And when that's, if this part is zero, then the f of five is Lowest value it can take on is zero cause you're squaring it, it can never take on a negative value. Going to hit a minimum point when this part of theĮxpression is equal to zero because this thing, the Realize, like, okay look, for this particular one, we're This is actually called vertex form because it's veryĮasy to pick out the vertex. And, when t is equal toįive, what is f of t? What is f of five? Well, when t is equal to five, five minus five squared is just zero. The t coordinate is five and five is three away from eightĪnd three away from two. It's going to be halfway in between where the parabola, in this case, is going to intersect the x axis, or the t axis, I keep saying x axis, the t axis for this case. Vertex is going to be halfway in between the zeros. T coordinate of the vertex since the input variable here is t, the t coordinate of the So, the x coordinate of the vertex, or sorry, I should say the T is equal to eight or two, the function is going to be zero. Have found the zeros for this function because if And to solve for t, we couldĪdd five to both sides, so we get t is equal toĮight or t is equal to, if we add five to both sides here, t is equal to two. Is nine, that means that t minus five could be equal to the positive square root of nine or t minus five could equal the negative square root of nine. We add nine to both sides the lefthand side's just Minus nine, equal zero? Let's see, to solve this, weĬould add nine to both sides. So, to find the zeros, we can set t minus five squared, So, let's see, so let'sįirst find the zeros. And if, at any point, you get inspired pause the video again and And I'm assuming you just did that and now I am going to attempt to do it. See if you can figure out the zeros, the vertex,Īnd the line of symmetry. Which will actually be the only line of symmetry for these three. And in particular, to make itĪ little bit more specific, the vertical line of symmetry, And finally, I want to find the equation of the line of symmetry. I also want to find theĬoordinates of the vertex. Here, it'd be the x values that make the function equal zero. And so, the zeros are the input values that make the value of theįunction equal to zero. I know they're all called f, but we're gonna just assume
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