The federal income tax laws are quite complicated. There have been various proposals to simplify the system. Many of these proposals are often referred to as a flat tax. To most people, a flat tax implies a single tax rate. This is usually not the case in most of the proposals. For example, a tax proposal endorsed by 1996 presidential candidate Steve Forbes proposed that a person would be taxed at a rate of 17% only on the income they earned above $13,300. The first $13,300 would not be taxed at all. 

1. Why is Steve Forbes flat tax function a piecewise function?
2. Give an equation for Steve Forbes flat tax where income is the input and the amount of the federal income tax is the output. What is the domain and range for your function?
3. How much would a person earning $20,000 per year pay in federal income tax? What percent of that person's income is used to pay federal income tax?
4. How much would a person earning $40,000 per year pay in federal income tax? What percent of that person's income is used to pay federal income tax?
5. You should see from your answers to questions 3 and 4 that the percent of income paid in federal income tax is not the same for everyone since you are only taxed on what you make above $13,300. Give a formula for the function where the input is the income of an individual and the output is the percent of that income that goes towards the federal income tax. What is the domain and range of this function?

You may have noticed that some transformations can either be a horizontal or a vertical  shift.  For example, the following graph shows f(x) = 1.5x - 6 and g(x) = 1.5x - 3. The graph of g can be considered a horizontal shift of f by moving it two units to the left or a vertical shift of  f by moving it three units up.  When can a transformation be obtained by either a vertical or horizontal shift? We will look at this question as it pertains to linear and exponential functions. 

1. Let's start by looking at vertical and horizontal shifts of linear equations. 
1. Let f(x) = 2x + 3. 
1. Determine the equation for y = f(x + 4). 
2. The horizontal shift of four units to the left for f is the same as a vertical shift of how many units and in what direction? 
2. Let f(x) = 2x + 3. 
1. Determine the equation for y = f(x - 4). 
2. The horizontal shift of four units to the right for f is the same as a vertical shift of how many units and in what direction?
3. Let g(x) = -x/3 + 5. 
1. Determine the equation for y = g(x - 2). 
2. The horizontal shift of two units to the right for g is the same as a vertical shift of how many units and in what direction? 
4. Let h(x) = mx + b. 
1. Determine the equation for y = h(x + c).
2. The horizontal shift of c units to the right for h is the same as a vertical shift of how many units and in what direction? 
 
2. Let's explore what type of vertical transformation is equivalent to a horizontal shift for an exponential function.
1. Let f(x) = 2^x.
1. Determine the equation for y = f(x + 3).  Simplify your equation so that the exponent is just x.
2. The horizontal shift of three to the left for f is the same as a vertical stretch by what factor? 
2. Let g(x)=0.4^x.
1. Determine the equation for y = g(x - 2).  Simplify your equation so that the exponent is just x.
2. The horizontal shift of two to the right for g is the same as a vertical stretch by what factor? 
3. Let h(x) = a^x, where a is a positive real number not equal to one.
1. Determine the equation for y = h(x + c) where c is some real number.  Simplify your equation so that the exponent is just x.
2. The horizontal shift of c units for h is the same as a vertical stretch by what factor? 

The historical note in this section mentioned that the Gateway Arch (part of the Jefferson National Expansion Memorial in St. Louis, Missouri) is not a parabola but is a shape known as a catenary. In this investigation, we will look at the shape of the Gateway Arch and compare it to a parabola. 

The catenary is the name given to the shape formed by the graph of the hyperbolic cosine (abbreviated cosh) and is the shape of a uniform flexible cable, or chain whose ends are supported from the same height. The word catenary comes from the Latin word for chain. The hyperbolic cosine is defined as 

The functions e^x and e^-x are exponential functions whose base is e, a number approximately equal to 2.718. The number e is an important constant that occurs in a variety of mathematical contexts and so is denoted by its own symbol (similar to the number p ). The exponential function y = e^x is often the "nicest" exponential function to use, partly because it crosses the y-axis with a slope of one. 
 

1. We want to examine the relationship between the graph of y = (1/2) e^x + (1/2) e^-x and the graphs of f(x) = (1/2) e^x and g(x) = (1/2) e^-x . 

1. On the same set of axes, sketch a graph of f(x) = (1/2) e^x and g(x) = (1/2) e^-x . 
2. WITHOUT sketching y = (1/2) e^x + (1/2) e^-x, predict the shape of the graph by looking at the graphs from part (a) and using your knowledge of what happens when you add functions together.
3. Graph y = (1/2) e^x + (1/2) e^-x. Was your prediction accurate?

2. Now let's look at the Gateway Arch. The equation that gives the shape of the Arch is 

1. Rewrite cosh 0.0100333x as a sum of exponential functions using the fact cosh x = 1/2 e^x + 1/2 e^-x. 
2. Rewrite the Gateway Arch function in terms of exponential functions instead of using the cosh x function.
3. The Arch has a height of approximately 625 feet and a span of approximately 600 feet. Using an appropriate viewing window, graph the function that gives the shape of the Gateway Arch.

3. We are interested in comparing the shape of the Gateway Arch to an appropriate parabola. 

1. The parabola we are going to use will be of the form y = -ax² + b.
1. Why will the value of a be negative?
2. Why will our parabola be of the form -ax² rather than -x²?
3. Why is it necessary to add b to the equation for our parabola?
2. We want a parabola that matches the Gateway Arch at the vertex and at ground level. This means our parabola must contain the points (0, 625), (-300, 0), and (300, 0). Using these points, find the values of a and b.
3. Graph the parabola on the same set of axes as the formula for the Gateway Arch.
4. How do your two graphs compare? How would you describe the shape of the Gateway Arch compared to the shape of a parabola?

Suppose a projectile was fired into the air at an angle of q . The vertical distance of the projectile (in feet) is given by v(t) = -16t² + v₀tsin q where t is time (in seconds) and v₀ is the initial velocity (in feet per second). The horizontal distance of the projectile (in feet) is given by h(t) = v₀tcos q . We are interested in determining the total horizontal distance (known as range) traveled by the projectile. 

1. Let r be the range of the projectile. Show that r = (1/32)v₀² sin 2q . [Hint: First use the vertical distance to find an expression for the time when the projectile has landed.]
2. Calculate the range of a projectile shot at an angle of 60° that has an initial velocity of 120 feet per second.
3. Calculate the range of a projectile shot at an angle of 30° that has an initial velocity of 120 feet per second.
4. Suppose the first projectile is shot at an angle of q and a second projectile is shot at an angle of 90° - q . Assume the initial velocity is the same for both projectiles. Which one travels farther? Justify your answer.
5. Suppose, given a fixed initial velocity, you want the projectile to travel as far as possible. At what angle should you fire the projectile? Justify your answer.
6. Suppose you have two projectiles, the first of which is shot at an angle of a and a second that is shot an angle of b. Assume the initial velocity is the same for both projectiles. If you want the first projectile to travel the farthest, what do you know about the relationship between a and b?