Part 1: Determining Paths of Miniature Golf Balls

(1)  The black dots in Handout 1 represent holes on a flat miniature golf course. The white circles are golf balls. Using Handout 1 and a ruler, determine a path for the golf ball that will result in a hole-in-one for each hole.

(2)  Explain your ‘shots’ in terms of equal angles, equal distances, and reflections.

(3)  What type of functions could be used to model the paths of the golf balls? Are the paths straight sections or curves? Could polynomials be used to model the paths?

(4)  Compare your paths with those of your classmates. As a class for each hole, choose one path and record it on Handout 2. You should keep track of the coordinates where the ball starts, where the ball hits a wall, and where the ball enters the hole. The axes are marked in units of 2 to match the graphs used by ExploreMath. 

(5)  Characterize the features of these paths. Compare and contrast them to linear and quadratic functions?  What is different about these graphs?

Part 2: Investigating Absolute Values of Linear Functions

Go to the “Absolute Value of Linear Functions” Activity. When the activity loads up, the following graph should be showing.

y = |f(x)| functions

(6)   On which intervals of f(x) = x + 2.0 are the values of y = f(x) positive?  Negative? Generate an (x, y) chart for f(x) using the graph to help determine the coordinates of f(x). Predict how the y values of      y = f(x) compare to the y values of y = |f(x)|.

(7)  Conjecture how the graph of y = f(x) is related to the graph of y = |f(x)|. What line of symmetry would y = |f(x)| have? Check your conjectures by selecting the black box next to y = |f(x)|.  

(8)  Manipulate m using the sliders. Observe changes in the graph. Now manipulate b using the sliders. Again, observe changes in the graph. Make general statements relating f(x) to |f(x)|.

y = f(|x|) functions

(9)  Manipulate the sliders to graph f(x) = x – 6 using the ExploreMath Activity. Using the graph fill in the table below for f(x) = x - 6.

(10)   Now make a table for f(|x|) = |x| - 6 for the same x values.

(11)  Conjecture how (x, y) is related to (-x, y) for f(|x|) = |x| - 6.  Does f(|x|) = |x| - 6 have a line of symmetry? What would be the line of symmetry?

(12)  Conjecture how the graph of f(x) = x – 6 can be used to aid in the graphing of f(|x|) = |x| - 6.  Check your conjectures by selecting the blue box next to y =  f(|x|).  The blue graph represents y = f(|x|).

(13)   Manipulate m and b using the sliders.  Make generalizations relating f(x) to f(|x|).

y = |f(|x|)| functions

(14)   Using the ExploreMath Activity, graph f(x) = 2x – 6 and f(|x|) = 2|x| - 6. On what intervals are the values of f(|x|)  = 2|x| - 6 negative?  Conjecture how the graph of |f(|x|)|  = |(2|x| - 6)| is different from the graph of f(|x|)  = 2|x| - 6 on that interval. 

(15)  On what intervals are the values of f(|x|)= 2|x| - 6 positive? Would the graph of |f(|x|)| = |(2|x| - 6)| be different than the graph of y = 2|x| - 6 on those intervals? Check the red box to see if your answers are correct.

(16)  Graph f(x) = -x + 2. Select the blue box to see f(|x|) = -|x| + 2 on the same axis. Using f(|x|) = -|x| + 2 as an aid, describe the graph of |f(|x|)| = |(-|x| + 2)|. Use the ExploreMath Activity to check your answers. Make generalizations relating the graph of f(x) to |f(|x|)|.

Part 3: Relating the Golf Ball Paths to Absolute Value Functions

(17)  Look at the graph of hole 1 on Handout 2. Consider the segment from the tee to the wall. Using the coordinates of the tee and the coordinates of the point where the ball hits the wall, find the equation of the line that contains this segment of the graph.

(18)  Experiment with the ExploreMath Activity to find the function that converts f(x) = x – 2 to the function that represents the path of the golf ball.

(19)  Look at hole 2 on Handout 2. The segment from the tee to the wall is a subsegment of the line f(x = x – 2, so experiment with the ExploreMath Activity to find the function that converts f(x)= x – 2 to the function that models the path of the golf ball.

(20)  Look at hole 3 on Handout 2. The segment from the tee to the first impact with the wall is a sub segment of f(x)= x – 2. Find the function that converts  f(x)= x – 2 to the function that models the path of the golf ball.

Part 4: Solving Absolute Value Equations

Inequalities of the form |x + b| > c

(21)  How could ExploreMath’s Activity be used to solve the inequality |x + 2| > 6

(22)  Graph |f(x)| = |x + 2|. For what x values does the graph have y values greater than 6? How do these intervals relate to the solution of |x + 2| > 6?

(23)  Manipulate the b slider. As b changes value, how do the intervals where |x + b|> 6 change? Conjecture about the solution of |x + b|> 6. How would the solutions to|x +b| < 6 compare to the solutions of |x + b| > 6?

Inequalities of the form |x| – b > 0

(24)  How could ExploreMath’s Activity be used to solve the inequality |x| – 4 > 0?

(25)  Graph f(x) = |x|-4. What are the x-intercepts of the graph? For what values does the graph of f(x)= |x| – 4 have y values greater than 0? How do these intervals relate to the solution of |x – 4|> 0?

(26)  Manipulate the b slider. As b changes value, how do the intervals where |x| - b > 0 change? Conjecture about the solution to the inequality of  |x| - b > 0. How would the solutions to |x| - b < 0 compare to the solutions of |x| - b > 0?

Extensions:

(1) Prove the hole-in-one paths work using geometric proofs.

(2)  Find other applications that can be modeled by functions involving absolute values.

(3)  Explore the solution sets for inequalities in the form |mx – b| > c and  |(|x| - b )| > 0.