Factored Form and Quadratic Graphs
October 15, 2026

[00:00:05] Is this a quadratic function? Yes. How can we tell? We multiply this out. The
[00:00:16] x times the x becomes the x squared. Definitely quadratic. You multiply any two linear, you
[00:00:21] get a quadratic. You multiply three linear, what do you get? So our goal is to graph this
[00:00:31] function just like we were graphing the quadratic functions. This is non-standard form. Is it
[00:00:40] in vertex form? No. Uh-oh. So how are we going to get it in one of those forms? We just got
[00:00:52] Multiply it out: x squared plus three x minus x minus three.
[00:01:16] Now it is in standard form. Standard form tells us two things. What are they?
[00:01:22] The y-intercept. We definitely know the y-intercept. And whether it
[00:01:35] opens up or opens down, which one is this? Open up because what's positive? Leading coefficient.
[00:01:47] Okay, so if we want to find
[00:01:49] vertex form, how are we going to do that? You could complete the square. The better way is to know the vertex is
[00:02:00] always at negative b over 2a comma f of negative b over 2a. Negative b over 2a is, what is b? Is 2. And a is 1.
[00:02:19] So that is negative one. That is the x-coordinate of the vertex.
[00:02:32] And how do we find the y coordinate? Plug it in to the function.
[00:02:36] Oh, no, that's right. I'm sorry. I got confused. Negative 4. We agree? Yes. Okay, sorry.
[00:02:45] So that is the y coordinate of the vertex.
[00:02:52] So what is our vertex form? The leading coefficient is one.
[00:03:05] Times x minus this number, which is x plus 1 squared minus 4.
[00:03:20] So the vertex (negative one, negative four) is right there. To graph it, follow the pattern: over one, up one;
[00:03:28] over two, up four.
[00:03:31] I'm just doing that because it's on the wrong side of the pattern.
[00:03:33] Over 1 up 1, over 2 up 1, 2, 3, 4. 1, 2, 3, up 9. Up here.
[00:03:44] That's going to be the right hand side. And how do we get the left hand side?
[00:03:47] On the other side. According to the axis of symmetry, which goes right through the vertex.
[00:03:57] And then we draw a nice curve.
[00:04:17] What are the x-intercepts?
[00:04:20] Negative three and one.
[00:04:24] Going back to the original quadratic.
[00:04:30] How can we look at that and tell the x-intercepts?
[00:04:37] Uh, we could use a zero product property.
[00:04:47] But it is simply the opposite of that number and the opposite of that number. Why is that?
[00:04:56] Yeah, right. We plug in the opposite of negative 1, which is 1.
[00:05:01] And you get 0 times whatever. It doesn't matter. It's 0.
[00:05:07] And the opposite of 3. So we plug in negative 3 plus 3. We get 0 times whatever. It's 0.
[00:05:16] Okay. So this is a special form.
[00:05:25] And it tells us something right away. What form is this? Oh, can anyone take this?
[00:05:31] Factored form, because the polynomial is factored.
[00:05:38] Okay. So this is called factored form.
[00:05:45] Okay. And what does it tell you? Those are the x-intercepts.
[00:05:59] Also known as zeros. I'll define that for you in a minute.
[00:06:09] Okay. If we have the x-intercepts
[00:06:20] without doing this negative b over 2a, how can we use the x-intercepts to figure out the x-coordinate
[00:06:26] of the vertex?
[00:06:28] We find the mean of these two?
[00:06:30] We find the mean, the average of the two, exactly halfway. Okay. Why is it exactly halfway?
[00:06:41] Because they are split about the axis of symmetry.
[00:06:46] Any two points on this parabola on the same horizontal line are symmetric.
[00:06:52] Symmetric ordered pairs. So if you find two such points,
[00:07:03] especially when they, like let's say they landed right here and here or something,
[00:07:08] where we couldn't know the zeros, we'll just go halfway. Exactly as the axis of symmetry,
[00:07:15] which is the x-coordinate of the vertex. So...
[00:07:22] So that's an alternative way,
[00:07:24] To find this value. And once we have that value, we just plug it into this function
[00:07:31] or this function to figure it out, the y-coordinate.
[00:07:38] So, right, if we plug in what's halfway? Negative one.
[00:07:43] Negative one minus one times negative one plus three, which is
[00:07:53] negative two times two, which is our negative four, our y-coordinate of the vertex.
[00:07:58] Okay, so that is factored form. So factored form
[00:08:17] Factored form is f of x equals the leading coefficient a, the same a as in vertex and standard forms, times x minus
[00:08:26] z one, times x minus z two, where
[00:08:36] z one and z two are the zeros, the x-coordinates of the intercepts.
[00:08:51] Okay, let me define the word zero for it.
[00:08:54] a zero
[00:09:00] of a function
[00:09:05] is an x value
[00:09:11] that makes
[00:09:16] the y-value, or function value,
[00:09:31] equal to zero, the number zero.
[00:09:44] It is the same as the x-coordinate of an x-intercept. Sometimes
[00:09:54] we could say the x-intercepts are negative three and one, or you can actually talk about the ordered pair.
[00:10:13] Typically, the x-intercept is actually an ordered pair, but we often just refer to it as a number.
[00:10:25] But these would be the zeros of the function. Zeros are numbers, x values that make the y value or
[00:10:31] the function value zero.
[00:10:32] What form is this in? Factored form.
[00:10:46] Okay, so what do we instantly know?
[00:10:49] What do we know from this form?
[00:10:52] The zeros.
[00:10:53] Zeros.
[00:10:55] Okay, also known as the x-intercepts.
[00:11:03] The zeros are the opposite of that number and the opposite of that number, two, negative four.
[00:11:16] Two, negative four.
[00:11:20] Okay.
[00:11:23] And then the vertex, the x-coordinate of the vertex,
[00:11:34] of the vertex is...
[00:11:42] Right, it's z one plus z two divided by two, which is two plus negative four divided by two.
[00:11:51] Here that is negative two over two, which is negative one.
[00:12:01] So we know this is the axis of symmetry.
[00:12:11] And then how would we find the y-coordinate of the vertex?
[00:12:17] Plug it in. We don't even need to multiply this out.
[00:12:24] So f of negative one is two times (negative one minus two) times (negative one plus four),
[00:12:32] which is two times negative three times three, which is
[00:12:44] negative eighteen. Uh-oh. So the only problem with this is way off my graph.
[00:12:52] Way down here at negative eighteen. Sorry, negative one comma negative eighteen.
[00:13:07] It's going to be a little harder to graph because I don't have a graph paper right here.
[00:13:19] But we would follow our pattern over... We're not going to go over one up one. Where are we going to go?
[00:13:28] Over one, up two. The parent offsets are (0,0), (1,1), (2,4), and (3,9).
[00:13:42] Our new pattern, instead of x and x squared, uses x and two x squared.
[00:13:51] Multiply those vertical offsets by two.
[00:14:01] Two, eight, eighteen.
[00:14:07] So I go over one up two.
[00:14:13] Over three up... Sorry, over two up eight.
[00:14:20] It's going to be right here at ten. I didn't draw this very well.
[00:14:25] Negative ten. And over three, up eighteen reaches the zeros right there.
[00:14:30] And then we got some symmetric points.
[00:14:37] We get a nice smooth curve.
[00:14:52] Okay.
[00:14:53] Okay. So what is the...
[00:15:01] What is the vertex form?
[00:15:09] Two...
[00:15:11] What was the vertex?
[00:15:20] Was it negative one...
[00:15:25] Negative eighteen.
[00:15:26] It is two times (x plus one) squared, minus eighteen.
[00:15:37] Okay. And then how would we get standard forms?
[00:15:45] We could either multiply this out, multiply it by two, and then subtract 18, or it's probably
[00:15:49] easier right here: two times (x squared plus four x minus two x minus eight).
[00:16:01] Which form should we... Let's go with the factored form first.
[00:16:15] Factored form is the leading coefficient times x minus one zero, times
[00:16:22] x minus the other zero. What are the zeros?
[00:16:30] Negative one, negative three.
[00:16:32] Negative one and negative three. So we have x minus negative one, or x plus one,
[00:16:39] and x minus negative three, or x plus three. How do we determine the leading coefficient from the graph?
[00:16:49] You start at the vertex. We go one to the right.
[00:16:55] And since we go down two, we know the leading coefficient is negative two.
[00:17:04] We can go right to the vertex form.
[00:17:15] The leading coefficient is negative two, times
[00:17:24] (x plus two) squared, plus two. Lots of twos.
[00:17:33] And then standard form. We know it's going to be
[00:17:40] a coefficient times x squared, plus or minus a coefficient times x.
[00:17:45] We do know the y-intercept, which is what? Negative six.
[00:17:50] But you need to multiply this out. So we got x squared
[00:17:57] negative two times x squared plus three x plus x plus three.
[00:18:09] Three x plus x is four x. Distributing negative two gives negative eight x and negative six.
[00:18:19] And that is standard form.
[00:18:21] What two things do we know?
[00:18:29] It opens upward because the leading coefficient is positive, and its y-intercept is negative one.
[00:18:40] How can we get this into factored form? One option is to factor it.
[00:18:50] Does it factor?
[00:18:54] The integer factors are one and negative one; those do not add to two.
[00:19:00] So how are we going to factor it?
[00:19:03] Find the zeros. We need to determine the zeros.
[00:19:11] How do we figure out the zeros
[00:19:18] Okay, we could do that, but I have a feeling
[00:19:26] They will not be obvious integers. This is just a rough sketch.
[00:19:33] Okay to find any zeros of a function
[00:19:37] You set the function equal to zero
[00:19:43] So we go zero equals x squared plus 2x minus 1
[00:19:54] And now we have a quadratic equation in one variable
[00:20:01] It cannot be solved by integer factoring. What is the next option?
[00:20:05] Use the quadratic formula.
[00:20:07] We sing it all together. Yes, we ready. Yes
[00:20:13] Negative b,
[00:20:16] the square root of b squared minus four a c.
[00:20:20] Negative two plus or minus the square root of two squared minus four times one times
[00:20:32] negative 1
[00:20:35] All over two.
[00:20:52] And
[00:20:53] The square root of eight is two square roots of two.
[00:21:05] Either factor out a two, or
[00:21:10] divide negative two by two and divide plus or minus two square roots of two by two.
[00:21:27] Those are the zeros, the x-coordinates of the intercepts.
[00:21:41] Oh
[00:21:41] So how do we put it in factored form?
[00:21:44] Well, do we know the leading coefficient which is
[00:21:48] one
[00:21:49] These are two numbers: negative one plus the square root of two
[00:21:53] and
[00:21:54] and negative one minus the square root of two.
[00:21:58] Factored form is
[00:22:01] One times x minus
[00:22:04] (negative one plus the square root of two),
[00:22:08] times x minus (negative one minus the square root of two).
[00:22:17] Distribute the signs to simplify.
[00:22:20] Get x
[00:22:23] plus one minus the square root of two;
[00:22:27] X plus 1
[00:22:30] Plus
[00:22:33] the square root of two.
[00:22:44] So
[00:22:45] So where the heck are those two numbers on the graph
[00:22:51] We'll get there in a second
[00:22:55] What is the square root of two approximately?
[00:23:00] One point something
[00:23:03] Let us use about
[00:23:05] One point four ish
[00:23:07] so negative one plus one point four and
[00:23:12] negative one minus one point four is
[00:23:16] Point four and negative two point four
[00:23:23] So point four and negative two point four
[00:23:37] Okay, the axis of symmetry is halfway
[00:23:43] Okay, which is the average of these two numbers
[00:23:48] Negative one plus the square root of two, plus negative one minus
[00:23:56] the square root of two, all over two.
[00:24:00] The two radical terms cancel to zero.
[00:24:05] So we get negative 2 over 2
[00:24:08] negative 1
[00:24:10] Okay, this is our axis of symmetry
[00:24:16] To get the y-coordinate of the vertex,
[00:24:19] evaluate f at negative one.
[00:24:23] F of negative one is (negative one) squared plus two times
[00:24:29] negative one,
[00:24:31] minus one,
[00:24:33] which is
[00:24:39] negative two.
[00:24:42] Follow our pattern
[00:24:58] Let's put it in vertex form
[00:25:03] The leading coefficient is one. What was the
[00:25:08] x-coordinate of the vertex? Negative one, so use (x plus one) squared,
[00:25:17] minus two.
[00:25:24] Okay
[00:25:26] So
[00:25:29] Almost every
[00:25:33] quadratic we have discussed can be written in all three forms. Each form tells us something different.
[00:25:40] We can convert to all the forms
[00:25:44] The only time
[00:25:47] We're not going to be able to write in factored form
[00:26:03] Okay
[00:26:05] Okay is this
[00:26:10] Why can we not write this in factored form
[00:26:14] If there are no x intercepts
[00:26:17] It cannot be factored into real linear factors.
[00:26:21] We know this equation, right?
[00:26:24] Or this function would be
[00:26:27] (x minus three) squared plus two.
[00:26:37] So how can you tell
[00:26:48] Yeah, if you have the graph it's obvious
[00:26:51] if the
[00:26:53] It opens up and above the x-axis
[00:26:55] It has no x-intercepts. Likewise, a downward-opening parabola with a negative maximum has no
[00:27:01] x-intercepts.
[00:27:03] What if we only have standard form?
[00:27:09] This is x squared minus 6x
[00:27:14] plus
[00:27:15] 11
[00:27:15] If we substitute into the
[00:27:20] Quadratic formula
[00:27:21] You're gonna get negative negative 6 plus or minus
[00:27:29] 36 minus 4 times 1 times 11
[00:27:36] Thirty-six minus forty-four. What is the radicand?
[00:27:43] Sorry, what's the discriminant?
[00:27:46] negative
[00:27:46] Negative eight, which means there are no real
[00:27:52] solutions to f of x equals zero: no real zeros and no x-intercepts.
[00:28:00] It cannot be written as a product of real linear factors.