Symmetry, Day 1

Today's lesson started with a review of identifying different types of symmetry in images. We looked at 17 different logos and determined if they have line symmetry or rotational symmetry.

For line symmetry, the image needs an imaginary line that presents mirror images. Many figures have vertical line symmetry, but the line can come in at any angle.

For rotational symmetry, the image needs to align with itself when it is spun around by less than 360° (every image has 360° rotational symmetry). To calculate the rotational symmetry, divide 360° by the number of points on the image that it can turn to.

For our graphs, we consider images that have vertical line symmetry at the y-axis to be even. And, images that have 180° rotational symmetry about the origin to be odd.

Homework: Worksheet on Graphs of Even & Odd Functions

Remember, there is a TEST scheduled for Thursday!

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Rationals - Day 1 (Thursday & Friday)

The last function that we will graph and translate this year is the rational function. This function is created when a variable is located in the denominator of a fraction. The parent function is

And its graph is

Notice that this graph is two separate curves. They do not connect in the middle. In fact, the center of the curves, (0, 0) in this graph, has two asymptotes running through it. An asymptote is a line on the graph that the curves do not cross, but instead they always approach. This means that the distance between the curves and the asymptotes is narrowing, but they do not intersect. On the graph of the parent function, the asymptotes are x = 0 (the y-axis) and y = 0 (the x-axis).

The graph of a rational function can be translate similar to our previous transformations. The general form of the rational function is

In this form, the graph translates according to:

• "a" stretches the graph if its absolute value is larger than 1;
• "a" compresses the graph if its absolute value is between 0 and 1;
• "a" reflects the graph upside down if it is a negative value;
• "h" shifts the graph to the left and right (remember, use the opposite of the sign that is "seen" in the function);
• "k" shifts the graph up and down.

Because of the nature of this graph, the characteristics of the graph are quite different from the characteristics of our previous functions.

• This graph has a center at (h, k) that is used to determine the asymptotes. The previous functions each have a vertex at (h, k).
• This graph has 2 asympotes: one vertical and one horizontal.
• This graph has a domain of all real numbers except the x-value of the vertical asymptote.
• This graph has a range of all real numbers except the y-value of the horizontal asymptote.
• This graph has end behaviors approaching the y-value of the horizontal asymptote.
• This graph is either increasing or decreasing, not both. And there are 2 separate intervals for the graph to be increasing (or decreasing).

Example: Graph the following function.

This graph has a center at (5, 3).

This graph has a vertical asymptote at x = 5.

This graph has a horizontal asymptote at y = 3

This graph is reflected upside down.

To find additional points for the graph, substitute in values for x around 5.

Homework: Students were to complete the notes that were handed out on Thursday. 1st period has already had this homework checked. 3rd and 4th periods will have this checked on Monday.

There will be a quiz on Thursday, March 25th on radical and rational functions.

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Radicals - Day 3 (Wednesday)

Sorry for the delay in posts. I was having trouble connecting to the blogging site from work. And while I can email blogs in, I have not figured out how to get the images to appear correctly when I do. Here is Wednesday's lesson from last week:

Today we brought together the previous two lessons of radicals to make a connection between the graphs and solutions of equations. In a radical equation, the two sides of the equation can be graphed as separate functions. Then the solution to the equation is located at the x-coordinate of the intersection of the two graphs.

Example: Solve the following equation by graphing.

Step #1: Break up the equation into two functions, one for each side of the equation:

and

Step #2: Graph the two equations.

Step #3: Find the intersection. For this graph, the functions intersect at (9, 6).



Step #4: Find the solution. The solution is the x-coordinate of the intersection. For this equation, x = 9.

We extended this out to finding what is called the zeroes of the function. The zero of a function is the solution of the function equal to 0. When graphing g(x) = 0, we find that it is the x-axis. So, solutions of the equation equal to 0, and zeroes of a function, and x-intercepts of a graph are all the same thing!



Homework: Worksheet on Solving and Graphing Radical Equations

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Radicals - Day 2

In class, students were reminded to take the online practice EOCTs every week. Many students currently have 2 zeros recorded for the last two weeks.

Today's lesson continued the work with radicals: we solved equations that contain radicals. To do this, follow these steps:

1. Isolate the square root by removing every value that is on the side with the square root. Remember, undo addition/subtraction first. Then undo multiplication/division.
2. Square both sides of the equation to undo the square root.
3. Solve the equation. Our equations today are so basic, this step is not needed. The equation will be solved in step #2.
4. Check the solution.

It is always important to check solutions for equations to verify that no mistakes were made. It is doubly important with radical equations because it is possible to accurately arrive at a "solution" that doesn't actually solve the equation. This occurs when step #1 sets up the square root to be equal to a negative number, which is not possible. However, if step #2 is completed, students will still arrive at a value that presents itself to be the solution. When the extraneous solution is substituted into the original equation, the student should arrive at a false equation (where the two sides are not equivalent).

Homework: Worksheet on Solving Radical Equations (found on the back of the notes)

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Radicals - Day 1

Today's lesson incorporated our graphic transformations (that we studied with the Quadratic and the Absolute Value function) with the square root function. The parent function looks like half of a parabola, turned on its side.

The generic form of the equation to graph a square root function is:

In this form, transformations are determined by:

• if the absolute value of a is greather than 1, then the graph is stretched taller;
  
• if the absolute value of a is between 0 and 1, then the graph is compressed shorter;
  
• if the value of a is less than 0, then the graph is reflected over the x-axis (flipped upside down);
  
• if the value of b is less than 0, then the graph is reflected over the y-axis (flipped over right to left);
  
• The value of h is always the opposite of whatever is "seen" within the square root (just like we did for the last 2 weeks) and this value shifts the graph to the left or right;
  
• The value of k is exactly whatever is "seen" after the square root (just like we did for the vertex form) and this value shift the graph up or down.

Students should always use a table of values to find points for their graphs. Pick x-values that will make the radicand (the inside of the square root) equal 0, equal 1, equal 4, and equal 9. We use these values because they are perfect square, so their square roots won't have us plotting decimal values for coordinates.

Example: Graph the following function:

• a = -2, so this graph will be flipped over and stretched taller by 2


• h = 1, so this graph will be shifted to the right 1 unit


• k = 4, so this graph will be shifted up 4 units


• use a table of to have points to graph: input x = -1, x = 2, x = 5, and x = 10.

Homework: Pages 140 & 141, #2 - 24 even

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Test day

Students are taking their test today on the quadratic concepts that we have been working with for the last 2 weeks.

This weekend is a math "holiday". Sunday's date is 3.14 which is also the common approximation used for pi. To celebrate this, students have been offered an extra credit opportunity to wear a shirt on Monday with some notes about pi on it. They received a handout with the information in class today.

Next week, we will start working with radicals (square roots) and rationals (fraction). Our next test is scheduled for April 1st.

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Review

We started class by going over homework. Then, student receive their Unit 4 Part 2 tests. For the most part, these grades were an improvement over previous tests. Students also receive a grade printout showing their current grade (with this most recent test). The statistics projects are not graded yet so those scores do not appear on the handout. Students should get their parents to sign off on the grade sheet.

Students received a review packet in class today. It is due tomorrow for a homework check and then we will go over it to prepare everyone for Thursday's test.

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