Unit+2+Virtual+Notebook

For the following three functions, complete a table of values for x values from -5 to 5 in your classroom binder and use your table of values to graph each function. You will find the three function in the document below. After evaluating for x values of -5 to 5 and graphing, answer the following questions in your virtual notebook. What type of function is f(x)? g(x)? and h(x)? Explain. f(x) is a parabola. It has an even exponent and the y-values are symmetrical. g(x) is a square root function. It's y-value never go below 0 and it's x-value never go beyond the x-value that has y-value of 0. h(x) is a odd function. It reflects over the x-axis. What observations did you make about the table of values and graph of f(x)? Explain how this relates to the function and why you think this happened. The graph of f(x) is symmetrical at the x value of 0. This relates to the function because its a parabola and parabolas have the same symmetry and distance between. What observations did you make about the table of values and graph of g(x)? Explain how this relates to the function and why you think this happened. The values of the negative g(x) come up as "error" on the calculator because it is impossible to have a negative number inside a square root sign. This relates to the function because it starts from -1 and looks similar to a linear line but it's slowly increasing. What observations did you make about the table of values and graph of h(x)? Explain how this relates to the function and why you think this happened.
 * __ Unit 2 Lesson 1 __**
 * Unit 2 Lesson 1**
 * < x ||< f(x) ||
 * < -5 ||< 20 ||
 * < -4 ||< 11 ||
 * < -3 ||< 4 ||
 * < -2 ||< -1 ||
 * < -1 ||< -4 ||
 * < 0 ||< -5 ||
 * < 1 ||< -4 ||
 * < 2 ||< -1 ||
 * < 3 ||< 4 ||
 * < 4 ||< 11 ||
 * < 5 ||< 20 ||
 * > x ||> g(x) ||
 * > -5 ||> --- ||
 * > -4 ||> --- ||
 * > -3 ||> --- ||
 * > -2 ||> --- ||
 * > -1 ||> 0 ||
 * > 0 ||> 1 ||
 * > 1 ||> 1.4142 ||
 * > 2 ||> 1.7321 ||
 * > 3 ||> 2 ||
 * > 4 ||> 2.2361 ||
 * > 5 ||> 2.4495 ||
 * x || h(x) ||
 * -5 || -0.125 ||
 * -4 || -0.1429 ||
 * -3 || -0.1667 ||
 * -2 || -0.2 ||
 * -1 || -0.25 ||
 * 0 || -0.3333 ||
 * 1 || -0.5 ||
 * 2 || -1 ||
 * 3 || --- ||
 * 4 || 1 ||
 * 5 || .5 ||

The values become smaller and smaller as it gets nearer to -3. This relates to the function because it's similar to a piecewise. The two lines on the graph are not connected because one goes the other way and vise versa. Look up the mathematical definition for domain and write what domain means in your own words. How do your observations made about each function and table of values relate to this definition? Explain.

A set of all possible input x values that are bounded. My observations of each function and table of values relate to this definition because the numbers that have a f(x), g(x) or h(x) are all possible but the ones with "error" are not possible or considered in the "domain" What do your think would be an appropriate domain for a function representing the population of deer from the years 1975-2005? Explain. 0<=x<=30 Its being assumed that its between 0 and 30 since that there are no other given value and that the difference between the two numbers are 30.

__** Unit 2 Lesson 2 **__



Find f(-3), f(1), when f(x) = 2, and f(x) = -2 f(-3)= 4 f(1)= -2.5 f(-1)= -2 f(-2)= 2 Where is this function increasing? (3, 5) Where is the function decreasing? (-3, -2) and (5, infinity) Where is the function constant? (-1, 3) How can you tell on a graph where a function is increasing, decreasing or constant? You can tell when it is increasing when the y-value increases and when is decreases when the y-value decreases. The graph can be determined constant if the y-value doesn't increase or decrease. Is the function continuous? Explain. Yes it is because there is no maximum or minimum. Find all local extrema of the function. What does it mean to ba a local maximum? What does it mean to be a local minimum? Can a function have more then one local maximum or minimum? Explain. Local Max.- infinity this is when the graph has reached its highest. in this case, infinity is the highest. Local min.- -infinity this is when the graph reaches its lowest, and that would be negative infinity.


 * __ Unit 2 Lesson 3 __**

The following graphs represent even and odd functions. I sorted the functions for you according to whether the function is an even function or an odd function.

__**Even Functions**__




 * __Odd Functions__**



In your virtual notebook answer the following questions:

Based on the classifications, when given a graphical representation what do you observe about all of the even functions? When given a graphical representation, it can be observed that all the graphs given are reflected over a common imaginary line. this line is the y axis. No matter how complicated the function is or how simple, it will always be symmetrical over the y axis. Based on the classifications, when given a graphical representation what do you observe about all of the odd functions? When given a graphical representation, it can be observed that all grpahs given are reflected over one common line. This line is the y=x line. It deos not matter how the complicated or simple the fucntion is, as long as it is symmetrical over the y=x line. Do you think a function always has to be odd or even? Explain. Support your answer with an example if necessary. A function doesn't always have to be odd or even. It could be neither since the graph might not be symmetrical over the y-axis or x-axis. It would be odd if it was symmetrical over the x-axis and even if it was over the y-axis. There are some functions that aren't symmetrical in anyway. How can you tell if a function is even or odd looking at a table of values? Explain. When it is odd, one side of the 0 should be all negative or positive numbers, yet on the other side, there shoud still be the same number, except with different positive/negative signs. Bottom line is, there should be opposite sings on either side of the 0, yet still have the same values. On the other hand, even numbers should have the same values and same postive/negative signs on both sides of the 0. How can you prove a function is even or odd algebraically? What steps should you take to prove whether a function is even of odd algebraically using the definition? Explain. Use the definition to find whether it is even or odd algebraically: Original- f(x) plug in both -f(x) and f(-x) When it is odd- plug in the definition and if the results are the same, then it is odd. When it is even- doing the same as the odd, plug in the equation and if it comes out as different answers from each other, then it is even.


 * __ Unit 2 Lesson 10 __**

1. In your own words, write the steps of performing a graphical transformation. Include any key reminders you think a students will forget in your description. 1. Horizontal Change 2. Reflection over the axis 3. Stretch or Shrink 4. Vertiacl Change 2. The graph of a function f(x) is illustrated. Use the graph of f(x) to perform the following graphical transformations. You do not need to show the shifted graph, you just need to list the 6 corresponding points.

(a) H(x) = f(x + 1) -2 (-1, 0) (0, -.2)  (1, -2)  (2, -1.5)  (3, -1)  (4, -.5) (b) Q(x) = 2f(x) (-2, 4) (-1, 3.6)  (0, 0)  (1, 1)  (2, 2)  (3, 3 ) (c) P(x) = -f(x) (-2, -2) (-1, -1.8)  (0, 0)  (1, -.5)  (2, -1)  (3, -1.5) 3. Suppose that the //x//-intercepts of the graph of f(x) are -5 and 3. Explain your thinking process or what helped you arrive at your answers. (a) What are //x//-intercepts if y = f(x+2)? x-int : (-2, 0) and (5, 0) (b) What are //x//-intercepts if y = f(x-2)? x-int : (-7, 0) and (1, 0) (c) What are //x//-intercepts if y = 4f(x)? x-int : (-5, 0) and (3, 0) (d) What are //x//-intercepts if y = f(-x)? x-int : (5, 0) and (-3, 0) 4. Suppose that the function f(x) is increasing on the interval (-1, 5). Explain your thinking process or what helped you arrive at your answers. (a) Over which interval is the graph of y = f(x+2) increasing? increase : (1, 7) (b) Over which interval is the graph of y = f(x-5) increasing? increase : (-6, 0) (c) Over which interval is the graph of y = f(x)-1 increasing? increase : (-1, 5) (d) Over which interval is the graph of y =- f(x) increasing? increase : (-1, 5) (e) Over which interval is the graph of y = f(-x) increasing? increase : (1, -5) 