WEBVTT
00:00:00.960 --> 00:00:07.210
In this video, we will learn how to determine the sign of a function from its equation or graph.
00:00:08.420 --> 00:00:12.630
We will begin by defining what we mean by the sign of a function.
00:00:13.730 --> 00:00:19.090
The sign of any function can be positive, negative, or equal to zero.
00:00:20.530 --> 00:00:26.790
If a function is positive, it is greater than zero and if it is negative, it is less than zero.
00:00:27.820 --> 00:00:30.570
Some functions might be more than one of these.
00:00:30.600 --> 00:00:36.640
They might be positive, negative, and equal to zero for different intervals of the function.
00:00:37.970 --> 00:00:40.890
Letβs consider some different types of graphs.
00:00:41.860 --> 00:00:45.340
We begin with three linear or straight-line graphs.
00:00:46.290 --> 00:00:52.620
The first graph is a horizontal line and will be of the form π¦ equals some constant π.
00:00:53.930 --> 00:00:59.140
As the line is below the π₯-axis, this function will always be negative.
00:01:00.850 --> 00:01:06.960
Our second graph is a vertical line, this time of the form π₯ equals some constant π.
00:01:08.090 --> 00:01:13.510
This function will be equal to zero at the point the line crosses the π₯-axis.
00:01:14.540 --> 00:01:20.300
It will be negative for all points below this and positive for all points above.
00:01:21.860 --> 00:01:32.860
Our third linear function is of the form π¦ equals ππ₯ plus π, where π is the slope or gradient and π is the π¦-intercept.
00:01:33.990 --> 00:01:38.880
Once again, this function will have one value where it is equal to zero.
00:01:40.280 --> 00:01:44.890
Part of the function will also be positive, and part of it will be negative.
00:01:45.300 --> 00:01:50.830
The bit above the π₯-axis is positive and the bit below is negative.
00:01:51.990 --> 00:01:58.650
We can calculate the value at which the function is equal to zero by setting π¦ equal to zero.
00:01:59.920 --> 00:02:04.140
We can then solve this equation to calculate a value of π₯.
00:02:04.980 --> 00:02:09.710
Letβs now consider what happens when we have a quadratic or cubic function.
00:02:10.800 --> 00:02:22.440
A quadratic function of the form ππ₯ squared plus ππ₯ plus π will be either u-shaped or n-shaped depending on the sign of the leading coefficient π.
00:02:23.810 --> 00:02:33.940
We will once again be able to calculate the values where the function is equal to zero by setting ππ₯ squared plus ππ₯ plus π equal to zero.
00:02:35.070 --> 00:02:42.910
In our diagram, the function will be positive to the right-hand side of one solution and to the left of the other.
00:02:44.090 --> 00:02:47.740
The function will be negative between the two values.
00:02:48.790 --> 00:02:58.440
Whilst it is not true for all quadratic functions, this function is positive, negative, and equal to zero for different values.
00:02:59.650 --> 00:03:03.040
The same is true for the cubic function shown.
00:03:04.270 --> 00:03:07.900
This is equal to zero at three points on the graph.
00:03:08.530 --> 00:03:14.900
It is positive between our first two solutions and when greater than the third solution.
00:03:15.890 --> 00:03:24.190
The function is negative when it is less than our first solution or between our second and third solutions.
00:03:25.880 --> 00:03:33.960
In this video, we will focus on questions involving constant functions, linear functions, and quadratic functions.
00:03:35.110 --> 00:03:40.490
In which of the following intervals is π of π₯ equal to negative eight negative?
00:03:41.370 --> 00:03:46.040
Is it (A) the open interval from negative β to eight?
00:03:46.740 --> 00:03:50.890
(B) The open interval from negative eight to β.
00:03:51.580 --> 00:03:55.570
(C) The open interval from negative eight to eight.
00:03:56.180 --> 00:03:59.940
(D) The open interval from eight to β.
00:04:00.450 --> 00:04:05.400
Or (E) the open interval from negative β to β.
00:04:07.220 --> 00:04:13.100
Letβs begin by considering what the function π of π₯ equals negative eight looks like.
00:04:13.990 --> 00:04:22.530
If we consider the normal coordinate axes, our horizontal axis is the π₯-axis and our vertical one is the π¦-axis.
00:04:23.200 --> 00:04:27.370
The π¦-axis can be replaced in this case with π of π₯.
00:04:28.430 --> 00:04:31.400
We are told that π of π₯ is equal to negative eight.
00:04:31.710 --> 00:04:36.660
Therefore, we need to find negative eight on the π¦- or π of π₯ axis.
00:04:37.780 --> 00:04:41.360
Our function is a horizontal line through this point.
00:04:42.380 --> 00:04:47.090
This line will continue indefinitely to the left and to the right.
00:04:47.930 --> 00:04:53.230
As the line is entirely below the π₯-axis, it will always be negative.
00:04:54.150 --> 00:05:04.880
As π of π₯ equal to negative eight is always negative, the correct answer is option (E) the open interval negative β to β.
00:05:05.830 --> 00:05:10.990
If any function is equal to a constant, it will always just have one sign.
00:05:11.380 --> 00:05:14.810
In this case, the function is always negative.
00:05:16.500 --> 00:05:22.510
In our next question, we will look at a function in the form π¦ equals ππ₯ plus π.
00:05:24.520 --> 00:05:30.630
Determine the sign of the function π of π₯ is equal to negative five π₯ plus five.
00:05:31.810 --> 00:05:43.740
We know that this function is linear as it is of the form π¦ equals ππ₯ plus π, where π is the gradient or slope and π is the π¦-intercept.
00:05:44.610 --> 00:05:51.200
In our question, the slope of the function is negative five and the π¦-intercept is five.
00:05:52.070 --> 00:05:57.920
As the slope of the function is negative, our graph will slope down to the right.
00:05:59.030 --> 00:06:08.220
In order to determine the sign of the function, we need to find out where the graph is positive, negative, and also equal to zero.
00:06:09.230 --> 00:06:12.900
We can see that the graph crosses the π₯-axis at one point.
00:06:13.280 --> 00:06:17.200
This will be the point where the function is equal to zero.
00:06:18.080 --> 00:06:27.070
When the graph is above the π₯-axis, the function will be positive, and when it is below the π₯-axis, it will be negative.
00:06:28.200 --> 00:06:34.910
To calculate the point at which the function is equal to zero, we will set π of π₯ equal to zero.
00:06:35.920 --> 00:06:41.300
Adding five π₯ to both sides of this equation gives us five π₯ is equal to five.
00:06:42.150 --> 00:06:47.930
We can then divide both sides of this equation by five, giving us π₯ is equal to one.
00:06:49.130 --> 00:06:54.110
Our function is positive for all π₯-values less than one.
00:06:55.110 --> 00:07:01.180
The function is negative or below the π₯-axis for all π₯-values greater than one.
00:07:02.020 --> 00:07:04.080
We can therefore conclude the following.
00:07:04.720 --> 00:07:18.550
The function is positive when π₯ is less than one, the function is negative when π₯ is greater than one, and, finally, the function equals zero when π₯ equals one.
00:07:19.670 --> 00:07:30.110
The function π of π₯ equals negative five π₯ plus five is positive, negative, and equals zero for different values of π₯.
00:07:32.220 --> 00:07:35.930
In our next question, we will look at a quadratic function.
00:07:37.470 --> 00:07:44.580
Determine the sign of the function π of π₯ is equal to π₯ squared plus 10π₯ plus 16.
00:07:45.740 --> 00:07:52.310
This function is quadratic and as the coefficient of π₯ squared is positive, it will be u-shaped.
00:07:53.330 --> 00:08:03.410
In order to determine the sign of any function, we need to work out values where π of π₯ is positive, negative, and also equal to zero.
00:08:04.440 --> 00:08:10.830
We can begin by calculating the zeroes of the function by setting π of π₯ equal to zero.
00:08:11.870 --> 00:08:17.060
This gives us π₯ squared plus 10π₯ plus 16 equals zero.
00:08:18.200 --> 00:08:23.350
The quadratic can be factored into two pairs of parentheses or brackets.
00:08:24.110 --> 00:08:31.200
The first term in each set of parentheses will be π₯ as π₯ multiplied by π₯ is π₯ squared.
00:08:32.400 --> 00:08:38.760
The second terms in our parentheses need to have a product of 16 and a sum of 10.
00:08:39.950 --> 00:08:45.660
Eight multiplied by two is equal to 16 and eight plus two is equal to 10.
00:08:46.660 --> 00:08:55.380
π₯ squared plus 10π₯ plus 16 factorized is equal to π₯ plus eight multiplied by π₯ plus two.
00:08:56.430 --> 00:09:04.240
As multiplying these two parentheses gives us zero, one of the parentheses themselves must also be equal to zero.
00:09:04.890 --> 00:09:09.840
Either π₯ plus eight equals zero or π₯ plus two equals zero.
00:09:10.810 --> 00:09:16.520
Subtracting eight from both sides of the first equation gives us π₯ equals negative eight.
00:09:17.500 --> 00:09:23.310
Subtracting two from both sides of the second equation gives us π₯ is equal to negative two.
00:09:24.710 --> 00:09:37.990
This means that the function π of π₯ equal to π₯ squared plus 10π₯ plus 16 is equal to zero when π₯ equals negative eight or π₯ equals negative two.
00:09:39.140 --> 00:09:45.440
It is now worth sketching the graph π¦ equals π₯ squared plus 10π₯ plus 16.
00:09:46.210 --> 00:09:54.820
We know that the graph is u-shaped and crosses the π₯-axis when π₯ equals negative eight and when π₯ equals negative two.
00:09:55.790 --> 00:10:00.540
We also know it crosses the π¦-axis when π¦ is equal to 16.
00:10:02.110 --> 00:10:06.010
The function is negative when it is below the π₯-axis.
00:10:06.470 --> 00:10:10.670
This occurs between the values negative eight and negative two.
00:10:11.610 --> 00:10:13.820
We can write this as an inequality.
00:10:14.290 --> 00:10:20.930
The function is negative when π₯ is greater than negative eight, but less than negative two.
00:10:21.930 --> 00:10:25.580
The graph is positive when it is above the π₯-axis.
00:10:25.960 --> 00:10:32.330
This occurs when π₯ is less than negative eight or when π₯ is greater than negative two.
00:10:33.380 --> 00:10:39.110
We can now write all of this information using interval and set notation.
00:10:40.120 --> 00:10:49.410
The function π of π₯ is positive for any real value apart from those in the closed interval negative eight to negative two.
00:10:50.530 --> 00:10:59.270
This means that the function is positive for all values apart from those between negative eight and negative two inclusive.
00:11:00.430 --> 00:11:07.180
The function is negative when π₯ exists in the open interval negative eight, negative two.
00:11:08.220 --> 00:11:16.650
This means that it is negative for any value between negative eight and negative two not including those values.
00:11:17.640 --> 00:11:25.070
The function is equal to zero when π₯ exists in the set of numbers negative eight, negative two.
00:11:26.300 --> 00:11:34.740
This means that it equals zero only at the two values π₯ equals negative eight and π₯ equals negative two.
00:11:35.960 --> 00:11:49.060
We can therefore see that the function π of π₯ is equal to π₯ squared plus 10π₯ plus 16 is positive, negative, and equal zero for different values of π₯.
00:11:50.520 --> 00:11:55.250
In our final question, we will consider two different functions.
00:11:57.060 --> 00:12:10.140
What are the values of π₯ for which the functions π of π₯ is equal to π₯ minus five and π of π₯ is equal to π₯ squared plus two π₯ minus 48 are both positive?
00:12:11.510 --> 00:12:16.660
Letβs begin by considering the function π of π₯ is equal to π₯ minus five.
00:12:17.410 --> 00:12:22.460
If we want this to be positive, π of π₯ must be greater than zero.
00:12:23.620 --> 00:12:27.680
This gives us π₯ minus five is greater than zero.
00:12:28.690 --> 00:12:34.580
Adding five to both sides of this inequality gives us π₯ is greater than five.
00:12:35.590 --> 00:12:41.300
π of π₯ is therefore positive on the open interval five to β.
00:12:41.760 --> 00:12:45.890
It is a positive function for any value greater than five.
00:12:47.230 --> 00:12:50.450
We will now repeat this process for π of π₯.
00:12:51.280 --> 00:12:57.440
This gives us π₯ squared plus two π₯ minus 48 is greater than zero.
00:12:58.390 --> 00:13:07.430
To solve any quadratic inequality of this form, we firstly need to find the zeros by setting our function equal to zero.
00:13:07.990 --> 00:13:12.250
π₯ squared plus two π₯ minus 48 equals zero.
00:13:13.110 --> 00:13:18.620
This can be factored or factorized into two sets of parentheses or brackets.
00:13:19.400 --> 00:13:22.170
The first term in each bracket is π₯.
00:13:23.110 --> 00:13:29.170
The second terms need to have a product of negative 48 and a sum of two.
00:13:30.160 --> 00:13:33.420
Six multiplied by eight is equal to 48.
00:13:34.540 --> 00:13:39.980
This means that negative six multiplied by eight is equal to negative 48.
00:13:40.780 --> 00:13:43.830
Negative six plus eight is equal to two.
00:13:44.780 --> 00:13:50.150
Our two sets of parentheses are π₯ minus six and π₯ plus eight.
00:13:51.110 --> 00:13:59.960
As the product of these two terms is equal to zero, either π₯ minus six equals zero or π₯ plus eight is equal to zero.
00:14:00.950 --> 00:14:05.950
Adding six to both sides of the first equation gives us π₯ is equal to six.
00:14:06.280 --> 00:14:12.320
And subtracting eight from both sides of the second equation gives us π₯ is equal to negative eight.
00:14:13.440 --> 00:14:21.170
This means that the function π of π₯ is equal to zero when π₯ equals six and π₯ equals negative eight.
00:14:22.290 --> 00:14:28.990
As our function is quadratic and the coefficient of π₯ squared is positive, the graph will be u-shaped.
00:14:29.910 --> 00:14:40.320
This means that it is positive on two sections, when π₯ is greater than six and when π₯ is less than negative eight.
00:14:41.190 --> 00:14:53.430
The solution to the inequality π₯ squared plus two π₯ minus 48 is greater than nought is π₯ is less than negative eight or π₯ is greater than six.
00:14:54.350 --> 00:14:57.530
This can also be written using interval notation.
00:14:57.970 --> 00:15:07.340
π of π₯ is positive in the open interval negative β to negative eight or the open interval six to β.
00:15:08.520 --> 00:15:13.180
We want to work out the values of π₯ where both functions are positive.
00:15:14.080 --> 00:15:20.510
Letβs consider a number line with the key values five, negative eight, and six marked on.
00:15:21.500 --> 00:15:26.670
We know that π of π₯ is positive for all values greater than five.
00:15:27.540 --> 00:15:34.460
π of π₯ is positive for all values less than negative eight and greater than six.
00:15:35.520 --> 00:15:40.940
This means that both functions are positive when π₯ is greater than six.
00:15:41.880 --> 00:15:48.640
This could also be written using interval notation as the open interval six to β.
00:15:50.620 --> 00:15:54.110
We will now summarize the key points from this video.
00:15:55.510 --> 00:16:03.530
A constant function of the form π of π₯ is equal to π will either be positive, negative, or equal to zero.
00:16:04.370 --> 00:16:08.380
If our value of π is positive, the function will be positive.
00:16:08.930 --> 00:16:11.270
When π is negative, it will be negative.
00:16:11.600 --> 00:16:16.390
And if π is equal to zero, the function will be equal to zero.
00:16:17.090 --> 00:16:28.050
A linear function of the form π of π₯ is equal to ππ₯ plus π will be positive, negative, and equal to zero for different values of π₯.
00:16:28.960 --> 00:16:34.380
We can find the value where it is equal to zero by setting π of π₯ equal to zero.
00:16:35.040 --> 00:16:42.310
It is then useful to draw the graph of the function to identify the points where it is negative and positive.
00:16:43.350 --> 00:16:54.870
A quadratic function of the form ππ₯ squared plus ππ₯ plus π is usually positive, negative, and equal to zero for different values of π₯.
00:16:55.990 --> 00:17:06.550
To work out the values where it is equal to zero, we once again set the function equal to zero and then factor the equation to calculate our values.
00:17:07.370 --> 00:17:15.390
Once weβve calculated these values, we can sketch the graph to find the points where the function is positive and negative.
00:17:16.300 --> 00:17:24.170
Whilst we didnβt see any questions of this type in the video, it is possible that the function has no values where it is equal to zero.
00:17:25.010 --> 00:17:31.930
In this case, the function would always be positive or always negative, as shown in the diagrams.
00:17:33.040 --> 00:17:40.070
For the vast majority of questions we see, however, there will be solutions where π of π₯ is equal to zero.
00:17:41.460 --> 00:17:47.060
If we canβt factor the function, we may still be able to solve it using the quadratic formula.
00:17:47.860 --> 00:17:56.150
We also saw that we can leave our answers to these type of questions using inequality signs or interval notation.