Unit+2

Unit 2 Lesson 1
What type of function is f(x)? g(x)? and h(x)? Explain. F(x) is a quadratic function. G(x) is a radical function. H(x) is a rational function.

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. In the table of values for f(x), they repeat themselves after 0. The graph has an axis of symmetry and the y values will be less than -5. This happened b/c it is a quadratic function, meaning that the x, whether positive or negative, will always turn out positive b/c it is squared. In this case, the corresponding negative and the positive values of x will have the same y term, but it appears that for all of the quadratic functions, there will be the same y value the same distance from the vertex.

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. In g(x), for all the x values less than -`1, they result in an imaginary #. The graph is can only be positive numbers. Since the function is a radical and it takes the square root of x+1, it makes sense that its domain can only be equal to or greater than -1 b/c anything less will make an imaginary number that cannot be shown on the graph. The domain seems that it has to be any number than will not make the expression under the radical a negative number.

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. For the table of h(x), there is a break at 3, since that makes the y value undefined. As the x values continue to be negative, the y values are smaller and smaller fractional negative values, and as the x values continue positive, it appears there will be the same trend (getting more fractional, but positive). The graph should have 2 branches that never their asymptotes, 0 for the horizontal, 3 for the vertical. This is b/c the x value can never be three, and since both sides are always becoming more fractional (because of the 1 is divided by the expression) when the x value is either increasing or decreasing, it’ll never reach 0 as a y term.

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 is the set of input values that fit the function. For f(x), you could see that the domain is all real numbers, since it fits the function. But for g(x), it has to be a set of numbers that make the expression under the radical positive, so the graph has a starting point. As for h(x), the domain is all real numbers except 3, since that makes the equation undefined.

What do you think would be an appropriate domain for a function representing the population of deer from the years 1975-2005? Explain. I think the domain for a function representing the population of deer should 30, since the interval b/w 1975 and 2005 is 30.

Unit 2, Lesson 2
Find f(-3), f(1), when f(x) = 2, and f(x) = -2 f(-3) = -2 and f(1) = -2

Where is this function increasing? (3, 5)

Where is the function decreasing? <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">(negative infinity, -1) (5, positive infinity)

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Where is the function constant? <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">(-1, 3)

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">How can you tell on a graph where a function is increasing, decreasing or constant? Can tell if increasing if the line is moving upwards, decreasing if downwards, and constant if straight horizontal line.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Is the function continuous? Explain. <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Yes, because there are no breaks or jumps.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">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. <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Local Max – 2

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Local Min – -2

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Local Max is the highest range value on some interval, and local min is the lowest range value on some interval. There can more than one local min or max b/c there might be more than 1 interval (of decreasing or increasing) on a single graph.

Unit 2, Lesson 3
<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Based on the classifications, when given a graphical representation what do you observe about all of the even functions? <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Even functions are symmetrical over the y axis.

Based on the classifications, when given a graphical representation what do you observe about all of the odd functions? <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Odd functions have origin symmetry.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Do you think a function always has to be odd or even? Explain. Support your answer with an example if necessary. <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">No, it doesn’t because there might be an equation like x^2 + x + 3, where the graph is moved diagonally across the graph, and it doesn’t touch the origin and it isn’t symmetrical over the y axis.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">How can you tell if a function is even or odd looking at a table of values? Explain. <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Could tell if a function is odd @ a table of values if the y values below and above 0 are the same, except they are opposite signs of each other. You can tell if a function is even when looking @ a table of values when the y values above and below 0 are exactly the same, with the same signs on either side.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">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. <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Could prove algebraically if it is an even function is f(-x) = f(x) and could prove algebraically if it is an odd function if f(-x) = -f(x). For an even function, would have to make all the x’s in the equation negative with parentheses and see if that simplified equation matches the original. For an odd function, would have to make all the x’s in the equation negative with parentheses and simplify as well. But then I would have to take the whole function itself and make it negative, simplifying it, and then seeing if that equation matched f(-x).

Unit 2, Lesson 10
First look for the h. When it is x-h, go h units to the right. When it is x+h, go h units to the left. Then, if the graph indicates it is reflected over the x or y axis, multiply the values of y or x by -1 respectively. After, if there is a number before the, it means to stretch or shrink, depending on the number, and you’ll have to multiply the y values by that number. Finally, look for the k value, which +k means go k units up, and –k means go k units down. Subtract from the y values depending on the k.


 * 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 separately.**

(a) h(x) = f(x + 1) -2 (left 1 and down 2 units) (-1, -2), (-7, -4), (-5, -2), (-3, 0), (3, 0), (5, 0)

(b) q(x) = 2f(x) (stretched by 2) (-6, -6), (-4, 0), (-2, 4), (0, 0), (4, 4), (6, 4)

(c) p(x) = -f(x) (reflection over x-axis) (-6, 3), (-4, 0), (-2, -2), (0,0), (4, -2), (6, -2) <span style="color: black; font-family: 'Times New Roman',serif;">
 * <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">3. Suppose that the ** //**<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">x **//**<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">-intercepts of the graph of f(x) are -5 and 3. Explain your thinking process or what helped you arrive at your answers. **

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> (a) What are //<span style="font-family: Arial,sans-serif;">x //-intercepts if y = f(x+2)? (shifted to the left two units) <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">-7 and 1 because h in this case is -2, so subtract 2 from each x value.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> (b) What are //<span style="font-family: Arial,sans-serif;">x //-intercepts if y = f(x-2)? (shifted to the right two units) <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">-3 and 5 because h in this case is +2, so add 2 to each x value.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> (c) What are //<span style="font-family: Arial,sans-serif;">x //-intercepts if y = 4f(x)? (stretched vertically by 4) <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Same, because stretching only affects the y values, and since the y values in this case are 0, anything multiplied by 0 is still 0, so -5 and 3 remain x-intercepts.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> (d) What are //<span style="font-family: Arial,sans-serif;">x //-intercepts if y = f(-x)? (reflected over the y-axis)

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> 5 and -3, because since you are reflecting over the y-axis, multiply all x values by -1.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> **<span style="font-family: Arial,sans-serif;">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. **

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> (a) Over which interval is the graph of y = f(x+2) increasing? (left 2 units) <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">(-3, 3) Since the increasing/decreasing intervals are represented by x values, and the graph shifted 2 units to the left, I subtracted 2 units from each x value

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> (b) Over which interval is the graph of y = f(x-5) increasing? (right 5 units) <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">(4, 10) The graph moved right 5 unites, affecting the x values, so I added 5 to each x value.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> (c) Over which interval is the graph of y = f(x)-1 increasing? (down 1 unit) <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Same, (-1, 5) Since increasing/decreasing intervals are represented by x values, and moving down 1 unit only affects the y values, the interval is the same s the original.

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> (d) Over which interval is the graph of y = -f(x) increasing? (reflected over x-axis) <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">Wouldn’t be increasing anymore. It would be decreasing if reflected over the y axis. Don’t know rest of what graph looks like so I can’t judge where it might be increasing anywhere else

<span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;"> (e) Over which interval is the graph of y = f(-x) increasing? (reflected over y-axis) <span style="color: black; font-family: Arial,sans-serif; font-size: 10pt;">(1, -5) Since it is reflected over y axis, multiply -1 to each value.

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