''AlgGeoStat - ** U3Q2 M Systems Take Home Quiz''
Write the inequality: Fix the inequality below to correctly show the domain of the function.
-1.5 greater than or equal to x greater than or equal to 1
2 x + y = 6
2 (0) + y = 6 0 + y = 6 y = 6 (0 , 6) = y-intercept 2x + 0 = 6 2 (3) 6 + 0 = 6 2x/2 = x , 6/2 = 3 x = 3 (3, 0) = x - intercept 2x + y = 6 - 2x -2x y = -2x + 6 - x + -2x + 6 = 3 -3x + 6 = 3 -6 -6 -3x/3 = -3/-3 x = 1 y = 4 ( 1 , 4)
Which of the following equations is written in standard form? 2x + y = 10 -2x + y = 12 y = 9x + 1 y - 4 = 7(x - 3)
2x + y = 10. This equation is written in standard form because the x and y are whole numbers and 2 is supposed to be positive.
What type of relation is a function?
A function is a type of relation where for each ___X, Y________, there is only one _INPUT, OUTPUT____________.
When is a linear function negative?
A function is negative when it is __BELOW_____________ the x-axis.
When is a function positive?
A function is positive when it is __ABOVE______________ the x-axis.
When is a linear function constant?
A linear function is constant when it has ___ZERO__________ slope.
Interval
An interval is a range of numbers. We write two values, separated by a comma, between brackets to represent all of the numbers between those values.
Standard:
Ax + By = C
Tails Enters the Race 0 yds 10 yds 20 yds 30 yds 40 yds 0 seconds Sonic's race is given by the function: f(x) = 15x−5 Tail's race is given by the function: f(x) = 10x+5 Your Solution: (2, 25)
Did the race match your graph? Why or Why not? It looks pretty close because Sonic seems to pass Tails after (2 seconds, 25 yds.)
What happened? Will Hanger B still balance? Why or why not?
Hanger B will still balance because it has the same quantity of squares total because both Hanger A and Hanger B have 4 sets of squares that are equivalent to the number 3.
Many Solutions
Lines on top of each other
Select an equation. Which equation(s) are parallel to the line shown here? (Select all that apply.)
y - 4 = - 5 (x - 2) y + 2 = -5 (x + 2)
Point-Slope Form
y - y1 = m ( x- x1) (x1 - y1) = any point on the graph M = slope
Mario Returns 0 yds 10 yds 20 yds 30 yds 40 yds 0 seconds Sonic's race is given by the function: f(x) = −15x+110 Tail's race is given by the function: f(x) = 12x+2 Your Solution: (4, 50)
Did the race match your graph? Why or Why not? Yes, the race did match our graph because both Sonic and Mario crossed paths when they were running in opposite directions after (4 seconds, at 50 yds.)
In this activity, you've made parallel lines by: • Dragging points with NO GRID • Dragging points WITH A GRID • Entering EQUATIONS What are the advantages of each approach?
Dragging points with NO GRID: We have to line it up more neatly. Dragging points WITH A GRID: We can notice that it has to have the same slope. Entering EQUATIONS: You know that it has to have the same slope, but different y-intercepts.
Exploring Range: Now we will identify the range of functions. The range is the set of y-values we get after substituting all the possible x-values into a function. Notice that the range of this function is y≤1, not y<1.
Even though there is not a closed dot placed at 2,1, EVERY point along the curve is a closed dot unless you actually see an open dot.
How can we determine if a graph represents a function or not?
Notice: We can determine if a graph represents a function or not by drawing a ______VERTICAL___________________ line. This is called the _____VERTICAL LINE TEST_______________________________________________________: If we draw a vertical line through a graph and it passes through ___TWO________ or more points, then the graph is not a function.
Journal Entry #9: What are two things you are looking forward to? Why?
One thing that I am looking forward to are the holidays/break to relax and try and take breaks from reality. Second, I am looking forward to the snow because I'll probably play outside with my siblings after school.
What does the m and b represent in slope-intercept form?
The m represents the slope. The b represents the y - intercept.
You may have noticed on previous screens that parallel lines have the SAME SLOPE. Here is an example of two such lines. What can you say about the solution (i.e., the point of intersection) to a system of two linear equations with the same slope?
The solution to a system of two linear equations with the same slope is going to have the same x-value, but not the same y-value.
When does a y-intercept occur? X-intercept?
A y-intercept occurs when ___x = 0_______________. An x-intercept occurs when _____y = 0_________________.
Find the Solution Drag the points to graph the system and find the solution: y=3x−1 y=−12x+6 Enter your solution in (x,y).
(2, 5)
Write the inequality: You haven't fixed the domain quite yet. That's okay. It's hard to get the domain exactly right by moving points. Mathematicians use inequalities to describe domain because they are more precise. Here is the inequality for the graph you created on screen 2. Fix the inequality to correctly represent the domain of the function.
1 greater than or equal to x greater than or equal to 4
How could you rewrite the blue equation with only x variables? y = 3x + 4 2x + y = 14
5x +4 = 14 (2, 10)
Hard Bracket
A hard bracket [ or ] is written if we want to ___INCLUDE_______ the number next to it in the interval. Example: Write an interval to represent the numbers between 3 and 5, including 3 and 5. [ 3, 5 ]
When is a linear function decreasing?
A linear function is decreasing when it has a ____NEGATIVE_____________ slope.
When is a linear function increasing?
A linear function is increasing when it has a ___POSITIVE_____________ slope.
Journal #8: Inspiration: Who inspires you? In what way? How? Describe why that person is inspiring to you and explain their qualities.
A person in my life that inspires me is my Grandpa because even though it is hard and he lives by himself he still takes care of himself and does the things that people should and are supposed to do like take walks and read and things like that. In addition, he is a very loving, kind, and caring person as well.
Relation:
A relation is a set of ordered pairs (x,y) where x is the input and y is the __OUTPUT________.
Soft Bracket
A soft bracket ( or ) is written if we __DON'T__________ want to include the number next to it in the interval. We always use a soft bracket with infinity and negative infinity. Example: Write an interval to represent the numbers greater than 2.
Mistake? Anton solved the following system using Substitution: x+3y=9 4x−5y=2 He got the solution −4.5,4.5 Ashley thinks he made a mistake. Who is correct?
Anton is correct. If he made a mistake, explain the mistake. If he is correct explain why. Anton is correct because with his equation he is trying to substitute the x in his solving.
Standard Form of a Linear Equation
Ax + By = C A, B, C constants (numbers) 4x + 3y = 12 * A and B are whole numbers A should be positive
Standard Form
Ax + By = C Constants: A, B, and C x and y are the variables
Function notation
Function notation helps us write the relationship between the inputs and outputs of a function. We will be able to write equations using function notation in this unit. Function notation is in the form f(a) = b, (read "f of a equals b") where a is the ___INPUT, X________ and b is the corresponding ___OUTPUT, Y_________. f (x) = y Example: f(3) = 2 ("f of 3 equals 2") means that when the input is 3, the output is 2. x = 3 y = 2
Function:
Function: A function is a type of relation where for each ____INPUT, X__________, there is only one ___OUTPUT, Y___________.
How did it go? How well did you understand the math today? How did you feel about learning in this lesson? It was ok 😳 I think I get it 😒 😎 Use the slides on the left. This is the math we wanted you to understand: I can find the solution to a system by graphing. What piece of information do you feel that you need more work on?
I need to work on graphing the lines from the last previous problems.
In 177 attempts, your classmates made perfectly parallel lines 1 time. Making parallel lines is difficult! What would make it easier?
I think the thing that would make it easier is a type of measuring device or resource to use.
Solving Systems of Equations: Solution
Intersect works in both equations
1 solution
Intersecting Lines
Tell A Story 0 yds 10 yds 20 yds 0 seconds Press Play. Tell a story, what is happening?
Mario starts at 5 yds and Sonic starts at 0 yds. and Sonic passes the finish line at 100 yds. (8.606 seconds). Then, Mario passes the finish line at 9.3 seconds.
Pause and Recap Messages encrypted with the Caesar cipher are very easy to crack, especially with a computational tool. Now that you've had a little practice cracking an alphabetic shift cipher (pretty easy, huh?) let's try something more difficult.
Recap terminology: Encryption - a process of encoding messages to keep them secret, so only "authorized" parties can read it. Decryption - a process that reverses encryption, taking a secret message and reproducing the original plain text Cipher - the generic term for a technique (or algorithm) that performs encryption Caesar's Cipher - a technique for encryption that shifts the alphabet by some number of characters. Cracking encryption - When you attempt to decode a secret message without knowing all the specifics of the cipher, you are trying to crack the encryption.
How do we write an equation in point-slope form?
So to write an equation in point-slope form, we need _______SLOPE________________________ and __ANY POINT ______________. NOTICE: Just like in the other forms of a linear equation, we should still have an x and a y in the final equation!
Domain Challenge #1: What is the domain of the function? Enter your answer in the box below. Be careful about ≤ vs. < and ≥ vs. >
The domain of the function is -4 greater than or equal to x and 4 greater than or equal to x.
Domain Challenge #2 The graph shows a point at ( minus 6 , minus 1 .678 ) and a point at ( 8 , 2 ), with a curve between these two points. Your inequality does not show the domain of the function. Please try again. What is the domain of the function? Enter your answer in the box below.
The domain of the function is -4 greater than or equal to x greater than or equal to 8.
What happened? How did the equation for Hanger B change?
The equation for Hanger B changed by the amount of 2 triangles or (2 x values) multiplying by 2 and the 5 purple circles or (5 y values) being substituted by those 4 blue triangles.
What are the two things we need to write an equation in point-slope form?
The two things we need for point-slope form are a slope and any point.
Exploring Domain: A mathematician says something is wrong with the shading on this graph. What do you think it is?
There are no points on the parallel shaded region and the line keeps continuing forever on the graph.
What do we call when we draw vertical lines through a graph to check whether it passes through 2 points to determine whether the graph is a function?
We call it the vertical line test because we're testing the vertical line and whether it determines that the graph is a function or not.
What do you notice about the graph?
We can tell Mario starts ahead and it's shown in the y-intercept. They're both at the same place at 4 seconds into the race. Also, you can tell that Sonic runs faster because he has a steeper slope as shown in the graph.
What does the Graph tell you?
What does the following tell you from the graph: Slope: 8.5 Y-Intercept: (0,10) Point (4,48)
Positive:
When is the graph above the x-axis?
Negative:
When is the graph below the x-axis? If you are talking about the left side of the graph we use (- infinity If you are talking about the right side of the graph we use (+infinity Your x intercepts should be used when talking about positive or negative
x-intercept:
Where does the graph cross the x-axis? Should be listed as (#, 0)
y-intercept:
Where does the graph cross the y-axis? Should be listed as (0, #)
Decreasing:
Where is the graph going down hill from left to right? ↘ Hint: if the whole thing is decreasing, you could say (- infinity, +infinity)
Increasing:
Where is the graph going up hill from left to right? ↗ Hint: if the whole thing is increasing, you could say (- infinity, +infinity)
Settle a dispute: Mark thinks the equation y=4+6x will match the red solid graph. Mia thinks it will match the blue dotted graph.
Who's right? Mia I think Mia is right because when I scanned the graph and found that the equation y = 4 + 6x was positive and matched the blue dotted graph because the y = 4. Also, you're adding the slope 1 space to the right and 6 spaces above, which matches Mia's answer.
Agree or disagree? Zeke claims that the function to the left should have three shaded regions for its range because it is a three-piece piecewise function. Is Zeke correct?
Yes Explain your thinking. Zeke is correct because there are three different pieces for a three-piece piece wise function. This is why it should have three different shaded regions.
How could you re-write the equation for Hanger B, using the information from Hanger A?
You could re-write the equation like this: 2x + 4y = 16.
Slope:
rise run = y2-y1x2-x1 You can also use the equation y=mx+b ; where m stands for slope
No Solution
two lines that are parallel and never intersect are said to have this
Slope Intercept:
y-intercept
Point Slope:
y-y1=m(x-x1)
Challenge #1: Adjust the equation so the line passes through the points.
y= 2/4 x
The equation of the DASHED line is y=−1.5x+3. Edit the equation of the SOLID line so that both the dashed and the solid line are EXACTLY parallel.
y=-1.5x-2
Solving Systems of Equations: Substitution - 2x + y = 6 -x + y = 3
y = -2x + 6
Desmos Example(s) - 2x + y = 4 2y = 4x - 4
y = 2x - 2 - 2x + y = 4 y = 2x - 2 2y = 4x - 4
Challenge #3: Adjust the equation so the line passes through the points.
y = 3/2x + 4
Slope Intercept Form
y = mx + b m = slope b = y-intercept
Strict or Non-Strict? Four possible answers for Screen 4 are below. Which best represents the domain shown to the left? You can adjust the inequality on Screen 4 if this helps with your thinking. −1.5<x<1 −1.5≤x<1 −1.5<x≤1 −1.5≤x≤1
-1.5 greater than or equal to x greater than or equal to 1 Explain your thinking. I chose this inequality because I know that the plotted point -1.5 is plotted on the left and x could be greater than or equal to either -1.5 or 1, but we do not know for sure because x does not have a value, and so this is why I chose this inequality.
Range Challenge #2 Fix the inequality below to correctly show the range of the function.
-4<x<-9
The lines shown here are ALMOST parallel, but not quite. Here are the equations of the lines: Dashed: y=−1.5x+3 Solid: y=−1.4x−2 1. What is ALMOST the same about the equations? 2. How could you change the equation of the solid line so that both the dashed and the solid line are EXACTLY parallel?
1. The the decimal for the slope of both 1.4 and 1.5 was off by .1 rather than having the exact slope, but they were really close in slope. 2. You could change the slope and change it to -1.5 to make it the same as the dashed line.
Can you solve the system without the model? Enter the solution to the system in (x,y). 3x = y 6x + y = 18
( 2, 6)
Sonic (Blue) runs back from the finish given by: f (x) = -14x+100 Mario (Red) runs from the start given by: f (x) = 11x Graph the two functions and find their solution in (x,y) form. Use the Zoom table to help you graph.
( 20/3 , 220/3)
Drag the points to graph the system and find the solution: y=−4x−1 y=x+4 Enter your solution in (x,y) and press submit.
(-1 , 3)
Find the Solution 2 Drag the points to graph the system and find the solution: y−6=34x−2 y−1=−23x−1 Enter your solution in (x,y).
(-2 , 3)
Drag the points to graph the system and find the solution: y−4=−13x+2 y+3=2x+2 Enter your solution in (x,y) and press submit.
(1,3)
Tails Enters the Race Sonic's (Blue) race is given by the function: f(x) = 15x-5 Tail's (Orange) race is given by the function: f (x) = 10x+5 Graph the two functions and find their solution in (x,y) form. Use the Zoom table to help you graph.
(2, 25)
Find the Solution 3 Drag the points to graph the system and find the solution: 3x+2y=13 −4x+5y=−2 Enter your solution in (x,y).
(3 , 2)
Sonic (Blue) begins 10 yards behind the starting line and run towards the finish at a rate of 15 yards per second. Mario (Red) starts 5 yards in front of the starting line and runs toward the finish at a rate of 10 yards per second. When and where will they meet? Graph the two functions and find their solution in (x, y) form. Use the Zoom table to help you graph.
(3, 35)
Mario Returns Sonic (Blue) runs back from 10 yards past the finish or 110 yards past the start of the race at a rate of 15 yards per second. Mario (Red) starts 2 yards in front of the starting line and runs toward the finish at a rate of 12 yards per second. When and where will they meet? Graph the two functions and find their solution in (x,y) form. Use the Zoom table to help you graph.
(4 , 50)
Cracking Substitution Ciphers The best technique for cracking a random substitution cipher is known as frequency analysis
Frequency analysis is a technique that is based on how frequently certain letters appear in English versus others. For instance, given a section of English text, E, T, A and O are the most common, while Z, Q and X are rare. Likewise, TH, ER, ON, and AN are the most common pairs of letters that occur next to each other. In fact, the distribution of letters is roughly the same for almost all samples of English text. The version of the widget on the previous page is intended to help you crack a substitution cipher through frequency analysis. By analyzing the frequency of the letters in the encrypted message compared to the frequency of letters in a typical piece of English prose, you can start to narrow in on what some of the letter mappings might be. The tool shows you how the frequency of letters in the encrypted text (orange) compares with frequencies from typical english (blue). Hint: Where to start? Find the short words and "crack" them first. How many one-letter words do you know? ("a"). A very common 3-letter word is "the". Once you've done that, you have substitions for some of the most common letters. You should be able to use intuition to look at other words with these partial subsititions and make good guesses. After finding only a handful of hard-fought letters, the rest will tumble quickly. Comparing the frequencies of letters gives good insight for making sensible guesses. Try this: The animation below shows someone getting started. Here's what they tried First sort the characters by frequency. Identify a group of characters that might map to the word the.It's a good start! Try this: The animation below shows someone getting started. Here's what they tried First sort the characters by frequency. Identify a group of characters that might map to the word the.It's a good start!
Favorite Form? Which form would you prefer to use to plot a line that passes through these points? Slope intercept form: y = mx + b
The form that I prefer to use to plot a line that passes through these points is Slope-intercept form because I'm very accustomed to doing it and that form is most relatable to me when it comes to math.
When can we use two different brackets?
We can use two different brackets if we want to include one number and not another. Example: (2,10] is an interval that DOES NOT include the number 2 but INCLUDES the number 10
Point-Slope Form
y - y1 = m(x - x1) m = slope y1 and x1 are the coordinates of a point on the line
Challenge #5: Adjust the equation so the line passes through the points. Hint: Adjust only the slope.
y = - 4/5 (x - 1) + 8
What is the equation of the line that is perpendicular to and has the same y-intercept as the graph shown?
y = -2 + 1/3 x Explain your thinking. I chose the equation y = -2 + 1/3x because the y-intercept was -2 on the graph and the slope of the line on the graph was 1/3.