1. Analyzing Graphs of Quadratic and Polynomial Functions

2. Vocabulary Domain: the set of x-values where the function is defined
Range: the set of y-values extracted from the function
Vertex: the maximum or minimum of a quadratic function
Local minimum: where the function has the lowest value in a certain region

3. More Vocab
Local maximum: where the function has the highest value in a certain region
x-intercept: where the graph crosses the x-axis;; y = 0; the solution to the function
y-intercept: where the graph crosses the y-axis; x = 0
Increasing interval: where the function is increasing from left to right
Decreasing interval: where the function is decreasing from left to right

4. Interval Notation
Instead of writing the intervals using inequalities, we can use interval notation. Click on the link to learn more about interval notation and how it compares to inequalities
Interval notation

5. If we wanted to write 4 < x ≤ 30 in interval notation, what would it look like?  
(4, 30]

6. Domain and Range
Remember, domain is all of your possible x-values and range is the y-values of the function

7. Maximum, Minimum, and x-intercepts
Refer back to the Mod 6 Lesson 1 notes as to how to find your max and min ordered pairs

8. Increasing and decreasing
Click on the link below and answer the questions on your notes sheet about increasing and decreasing intervals
Math is fun

9. Positive and negative
We can tell when the function has positive and negative values by its y-values.
Positive y-values – function is above the x-axis
Negative y-values – function is below the x-axis
You will need to find the x-intercepts of the function to help you identify these intervals

10. Symmetry There are many different ways a function can show symmetry.
Quadratic functions have an axis of symmetry – a vertical line that goes through the vertex.
It can be found by using the formula
It is the x-value of the vertex

11. Finding A.o.S. Find the axis of symmetry of the function
y = x2 – 2x + 5
= - (-2) = 1
2(1)
So x = 1 is the axis of symmetry

12. Other types of symmetry
Even: when the function is symmetric about the y-axis
Algebraically: f(-x) = f(x)
This means when you plug in a negative x-value, you get the same y-value as if you plugged in the positive x-value
Odd: when the function is symmetric about the origin
Algebraically: f(-x) = -f(x)
This means when you plug in a negative x-value, you get the opposite sign of the y-value as if you plugged in the positive x-value

13. Determine whether the function is even, odd, or neither
1. f(x) = x2 + 2 f(-x) = (-x)2 + 2 = x2 + 2 = f(x) therefore the function is even 2. f(x) = x4 – 2x + 5 f(-x) = (-x)4 – 2(-x) + 5 = x4 + 2x + 5 this is not f(x) nor –f(x) so this function is neither even nor odd

14. 3. f(x) = x5 + x3 - 3x f(-x) = (-x)5 + (-x)3 – 3(-x) = -x5 – x3 + 3x = - f(x) so the function is odd DO NOT assume you can tell even or odd by the degree of the polynomial.

15. Transformations Graph y = x2 and y = x2 + 2 on the same graph.
What do you notice?
Graph y = x2 and y = (x – 2)2 on the same graph.

16. The graph shifts up 2 units the graph shifts right 2 units

17. Transformations Graph y = x2 and y = 2x2 on the same graph.
What do you notice?
Graph y = x2 and y = -x2 on the same graph.

18. The graph is vertically stretched
The graph is vertically stretched. The graph is reflected over the x-axis.

19. Transformations
When we look at these transformations, we can see each piece shifts the graph in a special way.
a: vertically stretches or compresses the graph (a>1 stretch, 0<a<1 compress)
If a is negative, it reflects is over the x-axis
h: shifts the graph left or right (x-h right, x+h left)
k: shifts the graph up or down (+k up, -k down)

20. Let’s identify the transformations
Quadratic function Cubic function
Vertically compressed reflected over x-axis
Right 1 vertically stretched
left 2
down 8

21. Write the function given the transformations:
Quadratic function shifted 2 units right and 5 units up
Cubic function shifted 3 units left, 7 units down, and reflected about the x-axis

22. Lets look at an example
Lets look at the key features of the graph for y=x2-4x+3. So lets graph it on our calculator. We see it is a parabola that opens up. So we do not have local maximums and put none, and we do have a local minimum. We’ve got 2 x-intercepts we will need to find and a y-intercept. Lets first look at the domain. The domain is all of our x-values, so if we look at this graph, our x-values are on the x-axis – the horizontal axis, and if we follow this graph, it looks like is going to keep going up for ever and ever on both sides, which means I can plug in any x-value I want to and get a number back. So our domain is going to be all real numbers and if we are going to write it in interval notation it will be negative infinity to positive infinity. Now usually we want to use those sideways eights, but in wordpad, or notepad, we don’t have that. Now lets look at our range. The range are our y-values. And if we look here, we have a minimum y-value and then everything else is going up. We can see we have our graph matching up with y-values for ever and ever, we just have to figure out what this minimum y-value is. So lets go ahead and find what that local minimum. Hit second trace, number three and it asks us three questions. So here is where our vertex is, so we want to go left of it and hit enter. We want to go right of it and hit enter, and we can guess. So our minimum is (2, -1). Remember, our calculator does rounding sometimes, so when you see those or , that is just a rounding error. This means our lowest y-value is -1 and because we include that, we are going to use a bracket. So our y-values go from -1 to infinity. And we have a parentheses with the infinity. We have calculated our local minimum. Lets find our y-intercept because we do not need to do anything on the calculator to find the y-intercept. If we go back and look at the graph, our y-intercept we learned was that “c” value. So 3 is our y-intercept. If we write that as an ordered pair, x is zero, y is three. Now if we can factor this, then we can find our two zeros. This is factorable, so we could do it, but I did want to show you how to find this on the calculator. So lets hit 2nd trace, we are finding our zeros or solutions to our graph. Here is our first zero, so we want to go to the left of it and hit enter, and go to the right and hit enter. And we can come back and guess. Our first zero is x equals 1, and we can write that as an ordered pair (1, 0). Then we have a second solution, so we will do 2nd trace and find the zero again. Remember zeros and x-intercepts and solutions are all the same name for the same value - where it crosses the x-axis. Another name for it is root. So we will go to the left. We can’t go too far from it or we may cross over the zero and the calculator has issues with that. It does not know which zero to find. So our other zero is (3, 0). So we have our 2 x-intercepts, our y-intercept, domain and range and max and min, and now we are going to look at our increasing and decreasing. So if we look at this graph, we read these from left to right. So if I start following my graph, my graph is going down until the vertex. Which means this is the interval the graph is decreasing. We always write these intervals with the x-values, so our vertex minimum was x = 2. so the function is decreasing until we get to 2, which means we are going from negative infinity to two. So as long as my x-values are negative infinity coming to two my graph is decreasing. Then after two the graph is increasing, so from two to infinity we have our increasing interval. Our end behavior is what is happening to our graph at the end, as we go off to positive infinity and negative infinity on our graph. And we look here, they are both rising. Our end behavior for both sides is rises, or it is going to positive infinity on both sides. If you refer to your notes, we talked about them as rises. So on the left is rises, and on the right, it is also rises.

23. Example 2
Lets look at another example. We have a polynomial function y=x3-x2-3x+2. And we are going to find the key features of this graph also. So lets go ahead and graph it on our calculator. We have that nice polynomial shape, that classic cubic look. We are going to look at domain and range first. Like I said before, the domain is all of the possible x-values. So if we pick any x-value we want to, it should match up with our graph – it should have a y-value. Of course, this graph gets really big really fast so we don’t see a lot of the graph, but our domain is going to be all real numbers or negative infinity to positive infinity. And this it the interval notation I am showing you. For our range, it is all of our y-values, so we want to see if we can match up our graph to any y-value we pick. And some y-values have multiple places, and that is okay. Since this goes up forever and down forever, all of our y-values are going to get hit, so our range is also all real numbers, or negative infinity to positive infinity. Now we have a maximum and a minimum. Remember it is not the max of the whole graph, it is just the maximum of that little area there, so we call it a local max. We have a local max and a local min we will need to find. So we can do that. 2nd trace, we will do the maximum first. And again, we will have 3 questions the calculator is going to ask you and you will need to do what it says. Go to the left, go to the right, give yourself a little guess and get a value for your maximum. Now, before I told you that calculators do have some rounding problems, but this is a case where we are not going to round it. Our maximum is going to be (-.721, 3.268). We do not round to integers unless we see that or That is the only time that we round. So we are going to do the same thing, except we are going to find our minimum. So there is our local minimum. Go to the left, to the right, and give ourselves a little guess. It will come up with the minimum as being (1.387, ). In this case, we have three x-intercepts, so we have three solutions to this graph. We are going to have to find all three of them. We are going to use our zero. And we have to make sure we are aware that whatever zero we are looking at, that is the one we are going to the left of and to the right of. We do not want to stray too far from that, because it takes a while and our calculator can have some problems. So (-1.618, 0) is our first x-intercept. We are going to do the same thing for the other two. (.618, 0); (2, 0). Now, y-intercept, since this is a nice polynomial, we can go to y equals and look at the equation and we are looking for the constant. That will be our y-intercept. Since there is a plus two at the end, our y-intercept is 2. x is zero, y is two. Now we are going to look as see what intervals our function is increasing and decreasing. Now were are reading from left to right, our function is increasing until the maximum, then decreasing, then increasing again. So since we have already found our local maximum and minimum, we pretty much have our interval. We said our function, from left to right, is increasing first which means we are going from negative infinity to the x-value of the maximum which is Ad then we have a second interval where we are starting at the minimum and the going up. So the minimum x-value is and we are going from to infinity for our second increase interval. And we are decreasing in the middle – in between the maximum and minimum. Our decrease interval will have our maximum x-value to our minimum x-value Our end behavior, if we are looking at negative infinity, our graph is falling, so on the left falling and on the right, it is rising. Those are our key features.