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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Systems of Equations - NonLinear

We used the review of systems of linear equations on Friday to jump to systems of non-linear equations today. Our lesson has systems that are either 1 linear & 1 quadratic equation OR 2 quadratic equations.

Students should be able to find solutions on a graph. To do this, identify where the graphs meet. There could be no solutions (no intersection), 1 solution, 2 solutions, or infinitely many solutions (when the graphs overlap completely).

Students should also be able to find solutions by working with the equations. To do this follow these steps:

1. Confirm that both equations are solved for y. If not, get y by itself.

 

2. Substitute the first expression for y in the second equation. Solve for x. This can be done by either (1) setting the equation equal to 0, factoring, and solving for x from the factors; or (2), only when b = 0, set x² equal to the constant value and then square rooting both sides. Remember, when using method #2, the solution is the positive & negative of the value acquired through square-rooting.

 

3. Substitute the solved value(s) for x into either of the two equations and solve for y.

 

4. To check the solutions, substitute the ordered pairs into both equations to verify the values are correct.

Example:

Homework: Worksheet on Solving Systems of Non-Linear Equations.

There will be a TEST on Thursday covering all quadratic concepts from last week and this week. Students will receive a review worksheet tomorrow in class.

 

 

Systems of Equations - Linear

We discussed the online test that all students should have finished at least once by today. Students are encouraged to retest their *missed problems* to raise their score. I do not suggest that students completely retake a *new* test. They should build on the test that they already answered some of the problems correctly. I will record and only use the highest score that they achieve. At this time, I have not received a test score from every student. They have until midnight to complete a test and not receive a ZERO for this week's assignment.

Today's lesson is review of 8th grade concepts: solving systems of linear equations. Their handout has directions and examples for solving a system of linear equations with three different methods.

Solving a System by Graphing:

• Step 1: Graph each equation on the same coordinate plane.


• Step 2: Identify where the two graphs meet (if anywhere). The point where the graphs meet is a solution of the system. It is possible for the two lines to be parallel and never intersect. Parallel lines have no common points, so the system has no solution.


• Step 3: Check the solution by substituting the values of x and y into both equations. Verify that it solves both equations.


• Example:

Solving a System by Substituting:

• Step 1: Solve one of the equations for either of its variables.


• Step 2: Substitute the expression from step 1 into the other equation and solve for the other variable.


• Step 3: Substitute the solved value from step 2 into either of the original equations and solve for the remaining variable.


• Step 4: Check the solution by substituting the values of x and y into both equations. Verify that it solves both equations.


• Example:

Solving a System by Eliminating:

• Step 1: The goal is to make one of the variables have opposite values for its coefficients in the two equations (like -5x and +5x ... or +4y and -4y). If the equations do not already have this requirement, then select one of the variables to affect. Multiply everything in the equation to get the desired coefficient on the selected variable. You may need to multiply both equations by different values to meet this requirement.


• Step 2: Add the two equations together by collecting like terms. Some students remember this step from last year when they wanted to either get "0x" (zero-x, but called ox) or "+ 0y" (plus zero-y, but called toy). Solve for the remaining variable.


• Step 3: Substitute the solved value from step 2 into either of the original equations and solve for the remainging variable.


• Step 4: Check the solution by substituting the values for x and y into both equations. Verify that it solves both equations.
• Example: Note ... the step numbers in the example below are each one value higher than the way I numbered my steps above.
  
  
  

Homework: Worksheet on Systems of Linear Equations

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Quadratics, Day 3

Our third lesson with quadratics is our final version of graphing quadratics. Today we discussed the vertex form of the quadratic function: y = a (x - h) ² + k. This form makes is the easiest for identifying the vertex of the parabola.

• The vertex is located at (h, k). Students need to remember that they switch the sign of what the "see" inside the parenthesis in the equation for h. The value of k is exactly what they "see" to the right of the parenthesis in the equation.
• Use a table to select values near the vertex. Substitute these values in for x and solve for y. Get at least 5 points on the graph.
• Plot these points and draw a U-shaped curve.

Homework: Worksheet on Graphing Quadratics in Vertex Form

Don't forget to do your vocabulary and your online EOCT practice test. Both are due tomorrow!!!

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Delayed Start today

Due to the 2-hour delay today, Milton High School started with 3rd period classes. This means that I did not see my 1st period class.

My other IAA classes took a notebook quiz and a county wide Checkpoint Test. 1st period already took the notebook quiz last week. And, 1st period just will not take the checkpoint test due to our schedule. The Checkpoint Test is a diagnostic test to assess how much information from the entire year the IAA students know right now. The results of this test will NOT affect a student's grade in anyway. Its purpose is to show me what information I may need to reteach before our final exam.

Yesterday, students were given a handout explaining how to access a website (www.usatestprep.com) that has practice tests correlated to our final exam. They have a weekly assignment to take the "Large Test" for "Mathematics 1" each week by Friday.

There is no new homework today. If students did not finish yesterday's worksheet, then they have tonight to do it. Or, they can use tonight to take care of their weekly EOCT Practice Test assignment.

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Quadratics - Intercept form

Students found out yesterday that graphing quadratics in standard form does not lend itself well for locating the x-intercepts. Sometimes the graph crosses between two "nice" integers. Sometimes the graph crosses the x-axis a significant distance away from the vertex and requires extra work to locate. So, today's lesson is about the intercept form which identifies the x-intercepts very easily.

Intercept form: y = a (x - p) (x - q)

• To find the x-intercepts in this form, set the factors equal to 0 and solve for x: (p, 0) and (q, 0).
• To find the vertex and the axis of symmetry in this form, remember that the graph is symmetrical, meaning that the AOS will have to be in the middle of the intercepts. To find this location, average p and q. This calculates the x-value of the vertex. Substitute this into the equation and solve for y.
• Pick another value for x near the vertex and find it corresponding y value. Plot all points and draw a smooth U-shaped curve.

Homework: Worksheet on Graphing Quadratics in Intercept Form

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New Unit!

We started a new unit today. Unit 5 has us going back to algebra concepts so we will review concepts from the fall and build on them.

Today's concept is graphing quadratic equations given in standard form, y = ax² + bx + c. Much of this lesson is review. However, one part of the lesson is new:

• To find the vertex of a parabola, first find -b/2a. This value equals x for the vertex. Then, plug this value into the equation and solve for y. These values locate the vertex.

Once the vertex has been located, use a table or chart to organize your work picking values for x, substituting them in, and solving for y. Plot these points to find the shape of the parabola.

Along with graphing the quadratic equation, we expect students to be able to answer the following questions:

• Opens: the graph opens up when a > 0 and the graph opens down when a < 0 ;
• Vertex: the point found using the directions above;
• Min/Max: the vertex is a minimum when the graph opens up and the vertex is a maximum when the graph opens down;
• AOS: the axis of symmetry is the line "x = (-b/2a)";
• X-intercepts: the points where the graph crosses the x-axis. This can be none, one, or two points. When the graph crosses between two integers, simply state "between (m, 0) and (n, 0)" where m and n are the integers;
• Domain: the domain is the set of all possible x that can be used in the equation. For the quadratic functions it is {all real numbers};
• Range: the range of a quadratic function is based on the vertex and whether it is a maximum or a minimum. When the vertex is at the top of the graph (maximum), then the range is {y ≥ the y-value in the vertex}. When the vertex is at the bottom of the graph (minimum), then the range is {y ≤ the y-value in the vertex};
• End Behavior: the arrows at the ends of the graph of a quadratic function either both point up (going toward positive infinity or +∞) or both point down (going toward negative infinity or -∞);
• Intervals of Increasing and Decreasing: the intervals are determined by where your pencil draws downward (decreasing) and where it draws upward (increasing) when making the parabola. If the interval is to the left of the AOS, then write {x < the x-value in the vertex}. If the interval is to the right of the AOD, then write {x > the x-value int the vertex}.

Homework: Page 107, #24 - 30 even; Page 109, #16 - 26 even. For all graphs, also include x-intercepts, domain, range, end behavior, and intervals of increasing and decreasing.

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