Unit+2+Virtual+Notebook

Lesson 1


 * What type of function is f(x)? g(x)? and h(x)? Explain.
 * F(x) is a quadratic function, because it creates a parabola, g(x) is a square root function, because it has a square root, h(x) is an inverse function, because of the graph.
 * 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 y values of f(x) are the same on x values equidistant from the vertex of the graph. The graph is a U shape. This happened because both positives and negatives squared is a positive number, so the positives and negatives of a number that x could be results in the same answer, and therefore a U shape is created.
 * 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 y values are whole numbers after a pattern of adding 3,5,7,9, so to the x value that yielded a whole y value, starting from zero. The values below -1 are undefined, because you can't have a square root of a negative number. The graph of g(x) is a half of a parabola tilted to the side, facing positive infinity, with no negative x values below -1, and no negative y values. This pattern makes the function increase its y value less and less as the x values go up. This is because the square of numbers increases exponentially.
 * 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.
 * The graph is two curves facing opposite directions. The y value is undefined at 3, and the x values result in the positives and negatives of certain y values. The y value is undefined at 3 because it would result in the denominator equaling 0.
 * 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.
 * Domain: all the x values in a function. The observations relate to this definition because the graph's appearance and the table depended on the domain of the function.
 * What do your think would be an appropriate domain for a function representing the population of deer from the years 1975-2005? Explain.
 * A good domain would be from 0 to 30, with each representing the number of years from 1975, because there are 30 years between 1975-2005, and it wouldn't make sense to use negative numbers or any number greater than 30.
 * Domain: all the x values in a function. The observations relate to this definition because the graph's appearance and the table depended on the domain of the function.
 * What do your think would be an appropriate domain for a function representing the population of deer from the years 1975-2005? Explain.
 * A good domain would be from 0 to 30, with each representing the number of years from 1975, because there are 30 years between 1975-2005, and it wouldn't make sense to use negative numbers or any number greater than 30.
 * A good domain would be from 0 to 30, with each representing the number of years from 1975, because there are 30 years between 1975-2005, and it wouldn't make sense to use negative numbers or any number greater than 30.

Lesson 3
 * Based on the classifications, when given a graphical representation what do you observe about all of the even functions?
 * All the even functions are symmetrical when cut down the middle.
 * Based on the classifications, when given a graphical representation what do you observe about all of the odd functions?
 * All the odd functions are rotationally symmetrical 180 degrees counter clockwise.
 * Do you think a function always has to be odd or even? Explain. Support your answer with an example if necessary.
 * No; a function does not have to be odd or even. There are functions where the graph is not symmetrical at all, but goes all over the place. In this case, the function is neither odd nor even.
 * How can you tell if a function is even or odd looking at a table of values? Explain.
 * In an odd function, the positives and negative of each respective x value yields a y value that has opposite signs, while in an even function, the positives and negatives of each respective x value yield the same y value.
 * 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.
 * If f(x) = f(-x), then the function is even. If f(-x) = -f(x), then the function is odd. To prove whether a function is eve nor odd algebraically, plug in each function with the correct changes and compare them. If the two forms of the function equal each other, then the function is either even or odd, depending on which forms matched.
 * In an odd function, the positives and negative of each respective x value yields a y value that has opposite signs, while in an even function, the positives and negatives of each respective x value yield the same y value.
 * 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.
 * If f(x) = f(-x), then the function is even. If f(-x) = -f(x), then the function is odd. To prove whether a function is eve nor odd algebraically, plug in each function with the correct changes and compare them. If the two forms of the function equal each other, then the function is either even or odd, depending on which forms matched.
 * If f(x) = f(-x), then the function is even. If f(-x) = -f(x), then the function is odd. To prove whether a function is eve nor odd algebraically, plug in each function with the correct changes and compare them. If the two forms of the function equal each other, then the function is either even or odd, depending on which forms matched.

Lesson 10 To perform a graphical transformation, first find the parent function. Compare the parent function of the transformed one, and note any changes, ex. addition within parentheses, or multiplication of the value in parentheses. To begin the transformation, first check inside the parentheses. If x is subtracted by a number, move right that number or units, and if x is subtracted, move left that number of units. If x is multiplied by a number, then multiply the x value of the parent function by that number. Outside the parentheses, if the value is multiplied by a number, multiply the y value of the parent function by that number. Finally, if the function ends with addition or subtraction, move up or down that number of units, respectively.
 * 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. **


 * 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. Answer each part seperately. **

(a) H(x) = f(x + 1) -2 (left 1 and down 2 units) (-6,-3) -> (-7,-5), (-4,0) -> (-5,-2) , (-2,2) -> (-3,0) , (0,0) -> (-1,-2) , (4,2) -> (3,0) , (6,2) -> (5,0) (b) Q(x) = 2f(x) (stretched by 2) (-6,-3) -> (-6,-6), (-4,0) -> (-4,0) , (-2,2) -> (-2,4) , (0,0) -> (0,0) , (4,2) -> (4,4) , (6,2) -> (6,4) (c) P(x) = -f(x) (reflection over x-axis) (-6,-3) -> (-6,3), (-4,0) -> (-4,0) , (-2,2) -> (-2,-2) , (0,0) -> (0,0) , (4,2) -> (4,-2) , (6,2) -> (6,-2)


 * 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)? (shifted to the left two units) The new x-intercepts are -7 and 1, because the x values shifted two units left. (b) What are //x//-intercepts if y = f(x-2)? (shifted to the right two units) The new x-intercepts are -3 and 5, because the x values shifted two units right. (c) What are //x//-intercepts if y = 4f(x)? (stretched vertically by 4) The x-intercepts are -20 and 12, because the x values were multiplied by 4. (d) What are //x//-intercepts if y = f(-x)? (reflected over the y-axis) The x-intercepts are 5 and -3, because they were reflected over the y-axis, giving them opposite signs.
 * 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? (left 2 units) (-3,5), because the x value was shifted two units to the left. (b) Over which interval is the graph of y = f(x-5) increasing? (right 5 units) (4,5), because the x value was shifted 5 units to the right. (c) Over which interval is the graph of y = f(x)-1 increasing? (down 1 unit) (-1,4), because the y value was shifted one unit down. (d) Over which interval is the graph of y = -f(x) increasing? (reflected over x-axis) (-1,-5), because the interval was reflected over the y axis, giving the y value the opposite sign. (e) Over which interval is the graph of y = f(-x) increasing? (reflected over y-axis) (1,5), because the interval was reflected over the y-axis, giving the x value the opposite sign.