美国高中数学1-1 Linear & Quadratic Functions

美国高中数学1-1 Linear & Quadratic Functions Section 1-1: Points and Lines This section is a basic review of lines and points and their relationship to graphing. It is your job to know each of these terms, so get to it and memorize them!! Demo: Points in the xy plane! (Exploremath - requires Shockwave) Demo: Distance Formula! (Exploremath - requires Shockwave) Demo: 2x2 Linear systems! (Exploremath - requires Shockwave) 1) A solution of an equation is an ordered pair that makes the equation true. 2) The x - intercept is where the graph crosses the x-axis (also called zero, root or solution). The y - intercept is of course, where the graph crosses the y-axis. 3) A linear equation is in the form of Ax + By = C where A and B are not both zero! This is called the general form of a linear equation and graphs into a line. (amazing!) Sketching the graph of a line! One technique for graphing a line is to find the intercepts. Recall that it takes only two points to determine a line! To find the y-intercept let x = 0 and to find the x-intercept let y = 0! Sketch the graph of 2x + 5y = 10. Solution: Find the y-intercept by letting x = 0 2(0) + 5y = 10 5y = 10 y = 2 Therefore, the y-intercept is (0, 2) Find the x-intercept by letting y = 0 2x + 5(0) = 10 2x = 10 x = 5 Therefore, the x-intercept is (5, 0) Here is the graph!! Remember, there are three types of lines: vertical, horizontal and slanted! You need to know the equations of the horizontal and vertical. Study the following graphs: Note that the equation of a vertical line is x = constant(yellow on graph) and the equation of a horizontal line is y = constant(purple on graph) Intersection of lines There are three ways two lines can intersect in a plane. They can intersect exactly at one point, or they can be parallel, or they can have an infinite number of solutions (same line). To find the intersection, solve the equations simultaneously. (at the same time!) Example: Find the intersection point for the equations 2x + 5y = 10 x + y = 5 Using linear combinations (add/subtract method), multiply #2 by -2 -2x - 2y = -10 2x + 5y = 10 3y = 0 y = 0 To find x, replace y with 0 in either of the original equations. x + 0 = 5 x = 5 The intersection point is (5, 0) The graph is below: Important to remember When using this method, if both variables are eliminated, the result depends on the truth/falseness of the statement. If the resulting statement is true, this means the lines are the same and you have an infinite number of solutions. But, if the resulting statement is false, the lines have no common solution, they are parallel!! Distance and Midpoint Formulas Let A = (x, y), B = (x, y) and M be the midpoint 1122 of AB. Then: Example) Find the distance between (-1, 9) and (4, -3). Then find the midpoint. Distance = Midpoint = That does it for the first section. It is basic algebra review. You should have no problem with it!! On to the next section Section 1-2: Slopes of Lines Demo: Slope (Exploremath - requires Shockwave) Demo: Slope - Intercept Form (Exploremath - requires Shockwave) Demo: Point-Slope Form (Exploremath - requires Shockwave) As you recall from basic algebra, the slope of a non-vertical line measures the steepness of the line as related to the x-axis. If two points are on the same line then the definition of slope is as follows: Important facts about slope!! 1) Horizontal lines have a slope = 0. 2) Vertical lines have undefined slope because the denominator is zero. Demo: affect of Slope on a line (James Brennan) Sample graphs of slope! Note: The blue line has a positive slope because it goes uphill from left to right. The red line has a negative slope because it goes downhill from left to right. The purple line has an undefined slope because it is vertical. The green line has zero as the slope because it is horizontal. Important facts about slope to remember! 1) The slope of y = mx + b is m. m is always constant for a line. 2) Two nonvertical lines are parallel if and only if they have the same slope. 3) Two lines are perpendicular if and only if their slopes are negative reciprocals. Sample problems 1) Find the slope of the line thru (4,3) and (5,7). Solution: m = (7 - 3)/(5 - 4) = 4/1 = 4 Make sure you subtract in the same order. 2) Find the slope of the line thru (4,5), (4,3). Solution: m = (5 - 3)/(4 - 4) = 2/0 = undefined. The line is vertical 3) Find the slope of the line with the equation y = 4x + 5 Solution: It is in the correct form so the slope is m = 4 4) Find the slope of the line with the equation 4x + 2y = 8 Solution: The equation needs to be placed in the correct form. 2y = -4x + 8 y = -2x + 4 Therefore, the slope m = -2. The line is going downhill. 5) y = 4x + 7, y = 4x - 3, y = (-1/4)x + 8. Tell which are parallel and which are perpendicular. Solution: The first and second are parallel because they have the same slope, while the third equation is perpendicular to the first two because the slope is a negative reciprocal. Hope you were "speedy" in going over this section! Let's move on! Arriba! Arriba! Andale! Section 1-3: Finding Equations of Lines Linear equations can be written in many different ways. The following chart represents a few of the more useful methods: Form used for answers.. The form Ax + By = C A,B,C cannot be fractions Line has slope m and The slope-intercept form y = mx + b y-intercept b. Line has slope m and The point-slope form y-y = m(x-x) 11 contains (x, y) 11 Line has x-intercept a and The intercept form y-intercept b. Sample problems 1) Find an equation of the line with x-intercept 6 and y-intercept 3. Solution: Use the intercept method from above with a = 6 and b = 3. x/6 + y/3 = 1 multiply by 6 to eliminate the fractions x + 2y = 6 2) Find an equation of the line through (1, 5) and (5, -1). Solution: Use point-slope form from above and find the slope first. m = (5 - (-1))/(1 - 5) = 6/-4 = -3/2 Now use the point- slope form using one of the given points. It doesn't matter which one! (y - 5) = (-3/2)(x - 1) multiply by 2 to eliminate the fraction. 2y - 10 = -3x + 3 Put in general form 3x + 2y = 13 (Voila!) 3) Find an equation of the line with y-intercept -2 and parallel to the line 4x + y = 2. Solution: The equation has the same slope as the given line because it is parallel. Find the slope of this line first. 4x + y = 2 y = -4x + 2 The slope m = -4 Now use the slope-intercept method to find the equation. y = -4x - 2 4x + y = -2 4) Write an equation of the perpendicular bisector of the segment joining (-2,3) and (4, -5). Solution: We need to do two things. First find the midpoint and then find the slope of the line. Step 1: Use the midpoint formula to find the midpoint! This is the point that the perpendicular bisector goes through. Now find the slope. m = (-5 - 3)/(4 - (-2)) = -8/6 = -4/3 But the perpendicular slope is a negative reciprocal. So, m = 3/4 Now use the point-slope form and get: y - (-1) = (3/4)(x - 1) Multiply by 4 to eliminate the fraction. 4y + 4 = 3x - 3 Put in the general form: -3x + 4y = -7 or 3x - 4y = 7 (why?) This should give you enough to stay ahead of the road-runner!! Beep! Beep! (Is this the Acme web page or what!!) Section 1-4: Linear Functions - Modeling Try the quiz at the bottom of the page! go to quiz A function describes the way two quantities are related. y = 2x + 1 is a function as it describes how y relates to x. Every y value is found by doubling the x value and adding 1. Function notation replaces y with f(x) (pronounced "f of x") and is written f(x) = 2x + 1. This allows us to use different letters to represent different functions. A linear function is a function that has the form f(x) = mx + b. Some examples are given below: f(x) = 5x - 7 f is a linear function of x g(t) = .7t - 5 g is a linear function of t h(s) = -5s + 2 h is a linear function of s w is a linear function of x w(x) = 7 called a constant function Sample problem The junior class has paid $100 to a disc jockey for a dance. Tickets for the dance are $4 each. 1) Express the net income as a function of the number of tickets sold. Solution: Let I(n) = net income from n tickets sold. Then the income is found by subtracting the cost of the dj from the money earned. I(n) = 4n - 100 2) Graph the function Solution: n can only be a non-negative integer. The domain of the function is n > 0, where n is an integer. The graph looks as follows: Notice that they lose $100 if they sell no tickets. The break-even point is selling 25 tickets. Why? They make money selling more than 25 tickets. This graph represents a mathematical model which is describing a real world situation. Notice that the graph is made up of individual points. Graphs of this nature are called discrete functions. If we connect the dots and form a line it is a continuous function. This is sometimes given as the graph, but keep in mind the values must be integers! Suppose the phone company charges 20 cents the first minute and 10 cents for each additional minute rounded up to the next minute. (1.2 minutes would round to 2 full minutes). Graph a mathematical model of the cost of a call lasting t minutes. Time in minutes Cost in cents t C(t) 0 < t < 1 20 1 < t < 2 30 2 < t < 3 40 3 < t < 4 50 Look at the above graph. Does it model what is happening with the cost of a phone call? It seems to. Notice the open dots at the left end of each line segment. This indicates that at time t = 0, there is no charge because you haven't made a phone call. According to the graph, what is the charge at t = 1?It's 20 cents like it is supposed to be! A linear equation that would somewhat model this would be the equation C(t) = 10x + 20. The graph above is called a step function because it looks like a series of small steps! The linear graph looks as follows: If you use this graph as the model, be aware that the estimates are over the actual cost because from point to point the change is constant(slope)! Well pardner, it's time to mosey on to the next section!! On to complex numbers!! Imagine that! (Do you get the pun?) Section 1-5: Complex Numbers Recall from Algebra II that square roots of negative numbers are not real numbers. The uses of these types of numbers has become quite common. Therefore, we have the following basic definition of an imaginary unit! The powers of i form a cycle. They repeat the same 56pattern as above. Thus, i = i, i = -1 etc. To make it easy to find higher powers of i, simply divide the power by 4 and the remainder tells you which of the above to 43 Divide 43 by 4 and get a remainder use. Example: i 3of 3 thus it is equivalent to i = -i!! Study the sample problems below for the basic arithmetic with imaginary numbers! The above examples illustrate that to work with imaginary numbers, separate the imaginary part from the real part and simplify the real part. Complex numbers Any number in the form a + bi is an imaginary number. The a is the real part and b is the imaginary part. Adding complex numbers is easy. simply combine like terms!! (5 + 3i) + (4 + 6i) = 9 + 9i (3 + 2i) - (4 - 3i) = -1 + 5i (remember to switch signs when subtracting) Multiplying is also fairly easy. Since complex numbers are binomials, to multiply use the foil method. 2(2 + 3i)(1 + 2i) = 2 + 4i + 3i + 6i = (2 - 6) + (4i + 3i) = -4 + 7i 2remember i = -1 Recall that a + bi and a - bi are complex conjugates. Their sum and product will always be a real number!! You can use this property to rationalize the division of two complex numbers by multiplying top and bottom by the complex conjugate of the denominator!! Look at the example below: That's our simple review of the complex numbers. This topic should be easily remembered from last year! On to quadratics! Section 1-6: Solving Quadratic Equations Any equation in the form 2 ax + bx + c = 0 with a not equal 0 is called a quadratic equation! A value that satisfies this equation is called a root, zero or solution of the equation!! We have three methods we can use to solve these equations: 1) factoring (easy) 2) completing the square 3) quadratic formula Factoring review! Solve each by factoring 21) x + 4x -5 = 0 Solution: factor (x + 5)(x - 1) = 0 Therefore, x = -5 or x = 1 2) (3x - 2)((x + 4) = -11 2Solution: Foil first 3x + 10x - 8 = -11 2Put in correct form 3x + 10x + 3 = 0 factor (3x + 1)(x + 3) = 0 Solution: x = -1/3 or x = -3 Completing the square!! Follow the explanation and sample problem to review completing the square 1) Use completing the square to find the solutions for: 22x - 12x - 9 = 0 Solution: Move the constant to the other 2side: 2x - 12x = 9 22Divide by the coefficient of x x - 6x = 9/2 2Take half the coefficient of x and square: x - 6x + 9 = 9/2 + 9 Factor the trinomial square: (x 2- 3) = 27/2 Take the square root of both sides: Simplify the radical: Isolate for x: Quadratic Formula As proved in class the quadratice formula is derived by completing the square. Here is the formula: 2If ax + bx + c = 0 then the roots of the equation are: Look familiar? It better!! Solve the following problem by using the quadratic formula. 22x + 5 = 3x 2 2x - 3x + 5 = 0 (putting in correct form) a = 2, b = -3 and c = 5 Use the formula: The Discriminant! The quantity under the radical in the quadratic formula can tell us alot about the nature of the solutions. Therefore, it is given a special name. The 2discriminant is b - 4ac If the discriminant is less than zero, then you will be taking the square root of a negative number yielding complex solutions. If the discriminant equals zero, you have one real solution (namely -b/2a) (Where have I heard that before?). If the discriminant is greater than zero, then we have two different real solutions. This is summarized in the following chart: Types of solutions discriminant 2b - 4ac < 0 Two complex conjugates 2b -4ac = 0 One real (double root) 2b - 4ac > 0 Two different real roots Warning! Warning! Danger Ahead!! Be extremely careful in solving problems like this: 2x = x It is very tempting to divide both sides by x to get: x = 1 This is incorrect, because you have completely eliminated one of the solutions. Never divide both sides of an equation by a variable if it cancels from both sides!! Correct way to do the problem is as follows: 2x = x 2x - x = 0 x(x - 1) = 0 Solutions are x = 0 and x = 1 Honk! Honk! That about does it for this section! Onward and upward! Section 1-7: Quadratic Functions - Graphs 2A quadratic function is in the form f(x) = ax + bx + c. A quadratic function graphs into a parabola, a curve shape like the McDonald's arches. Every quadratic function has a vertex and a vertical axis of symmetry. This means it is the same graph on either side of a vertical line. The axis of symmetry goes through the vertex point. The vertex point is either the maximum or minimum point for the parabola. A summary of what we talked about last year is shown below: a>0 graph opens up a<0 graph opens down 2b - 4ac > 0 Graph has 2 x-intercepts 2b - 4ac = 0 Graph has 1 x-intercept 2b - 4ac < 0 Graph has no x-intercepts at x = -b/2a Vertex Point to find y substitute above value for x x-intercepts factor or use quadratic formula 2y-intercept c value in ax + bx + c Sample Problems Find the vertex point, axis of symmetry, x and y intercepts and graph. 21) F(x) = x + 2x - 3 Solution: The graph opens up because a is positive. The vertex point is at x = -b/2a = -2/2 = -1 2y = (-1) + 2(-1) - 3 = -4 Vertex point is at (-1, -4) Axis of symmetry is the equation x = -1 x-intercepts found by factoring (x + 3)(x - 1) = 0 They are at x = -3, x = 1 y-intercept at (0, c) at (0, -3) The graph is as follows: Alternate method! A second form of a quadratic can be written in the following form: 2f(x) = a(x - h) + k Where (h, k) is the vertex point Example Find the vertex point using the alternate method. 21) y = 3(x - 3) + 4 Solution: The vertex point is (3, 4) 22) y = 2(x + 1) + 3 Solution: The vertex point is (-1, 3) 23) y = 5(x + 2) - 7 Solution: The vertex point is (-2, -7) 24) y = 2x + 8x - 5 Solution: Put the equation in the correct form: 2y = 2(x + 4x + __ ) - 5 2y = 2(x +4x + 4) -5 - 8(completing the square) 2y = 2(x + 2) - 13 Therefore, the vertex point is (-2, -13) 2nd Sample Problem! Use the second method to graph the following equation! 21) y = x - 6x -7 Solution: Graph opens up because a is positive x-intercepts found by factoring (x - 7)(x + 1) = 0 The x-intercepts are x = 7, and x = -1 y-intercept is at (0,c) or (0, -7) Using method two to find the vertex point 2y = (x - 6x + ___) - 7 2y = (x - 6x + 9) - 7 - 9 2y = (x - 3) - 16 The vertex point is (3, -16) The axis of symmetry is at x = 3 The graph looks like this: That concludes our review of quadratics and graphing. On to quadratic modeling!! Section 1-8: Quadratic Models Try the quiz at the bottom of the page! go to quiz Quadratic models are used to model certain real-world situations such as: 1) Values decrease then increase 2) Values increase and then decrease 3) Values depending on surface area. 4) Objects thrown into the air. Application of curve fitting Suppose a function has the following points f(0) = 5, f(1) = 10 and f(2) = 19. Find an equation of the form f(x) = 2ax + bx + c. Solution: plug in each point to get three equations 5 = 0 + 0 + c means c = 5 10 = a + b + 5 means a + b = 5 19 = 4a + 2b + 5 means 4a + 2b = 14 The last two equations can be solved simultaneously a + b = 5 2a + b = 7 Subtract the first equation from the second to get: a = 2 This means that b = 3 2Therefore, the equation is 2x + 3x + 5 = f(x) Application to physics! An object thrown into the air is modeled by the equation 2h(t) = -4.9t + vt + h oo Where v is the initial velocity o h is the initial height above the ground o h(t) is the height after t seconds. A baseball is thrown with an upward velocity of 14 m/sec from a building 30m high. 1) Find its height above the ground t seconds later. 2Solution: h(t) = -4.9t + 14 t + 30 Simply substitute in the values. 2) When will the ball reach its peak? Solution: Find its vertex point. x = -b/2a = -14/-9.8 = 140/98 = 10/7 seconds later. 3) When will the ball hit the ground? Solution: This will happen when the height is zero. 20 = -4.9x + 14t + 30 249t - 140t -300 = 0 (7x + 10)(7x - 30) = 0 x = -10/7 or x = 30/7 Throw out -10/7 Why? Answer is 30/7 seconds. On to the sample test!! Good luck! Current quizaroo # 1b 1) Simplify (5 -3i)(2 + 5i) a) -5 + 19i b) -5 + 31i c) 25+ 19i d) 25 + 31i e) 25 - 19i 2) Write in the form a + bi, the following division problem 1/(3 + 2i) a) (3/5) - (2/5)i b) (3/13) - (2/13)i c) (3/5) + (2/5)i d) (3/13) + (2/13)i e) 3 - 2i 3) Solve the quadratic formula by any method: (2x + 1)(4x - 3) = (4x - 23) a) 4/3, -1 b) 0, -1 c) -4/3, 1 d) -3/4, 1 e) 3/4, 2 4) Name in order the vertex point, axis of symmetry, x-intercepts and y-intercepts for: 2 y = x + 4x + 3 a) (-2,-1), x = -2, (-1, 0) and (-3, 0), (0, 3) b) (2, 11), x = 2, (-1, 0) and (-3, 0), (0, 3) c) (-2, 1), x = 2, (1, 0) and (3, 0), (0, 3) d) (2, 11), x = -2, (1, 0) and (-3, 0), (0, 3) e) (-4, 3), x = 2, (-1, 0) and (-3, 0), (0, 3) 5) If you drive at x miles per hour and apply your brakes, your 2stopping distance in feet is approximately f(x) = x + (x/25). By how much does your stopping distance increase if you increase your speed from 30 to 40 mi/h? a) 104 ft b) 66 ft c) 38 ft d) 170 ft e) doesn't change click here for answers!! Current quizaroo # 1a 1) Find the midpoint of these two coordinates (1, 5) and ( -3, -9) a) (-1, -2) b) (2, 7) c) (1, 2) d) (4, 14) e) (-2, -4) 2) Find the slope of the lines to tell what is happening with the lines: 2x + 3y =6 and y = (-2/3)x + 2 a) Lines are parallel b) Lines are perpendicular c) Lines intersect but not at right angles d) Lines coincide e) Not enough info to tell 3) Write the equation of the line with slope = -2 and y-intercept = 5 a) 5x + y = 2 b) 5x + y = -2 c) 2x + y = -5 d) -2x + y = 5 e) 2x + y = 5 4) Find the equation of the line perpendicular to the line y = (1/2)x + 4 going through the point (0,2) a) y = 2x b) y = -2x c) y = 2x - 2 d) y = -2x + 2 e) 2y = x + 4 5) Jack's new truck costs $325 per month for car payments. He estimates that gas and maintenance expenses cost $.23 per mile. Express the monthly cost as a function of the miles driven. a) C(m) = 325m + .23 b) C(m) = -.23m + 325 c) C(m) = .23m + 325 d) C(m) = .23m - 325 e) C(m) = 325m - .23 click here for answers!! Quiz Answers Quiz # Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 1- A A D E D C 1 - B C B E A C 2 - A D A C B E 2 - B B C D A B 3 C A B E C 4 - A D B A D B 4 - B D A B A C 5 - A C A B E B 5 - B C D E B C 6 B D A C C 7 A B D B A 8 B C A C E 9 B A C E B 10 B C B A D 11 D C B A B 13 - A B A C E C 13 - B D B A B C 19 - A 1/9 -14 -1/4 - INFINITY INFINITY 19 - B -2 -1/2 X = 4 Y = 1 (-3, 1/7) 20 - A 2x + 4 + h 8 6x^-1/3 -12/x^4 8 20 - B (1,0) & (-2,0) (-2,0) & (0, -4) (-2,0) (0, -4) NONE CALC B D C A D Sample Test 1) Find the distance and midpoint for the points (1, 7) and ( 5, -3). 2) Find the intersection of the lines x + y = 5 and 2x - y = 1. Then graph the lines. 3) Find the slopes of the following: a) (5, 3) and (2, -3) b) 4x + 5y = 10 c) y = -5 4) Show that the line through (2, -3) and (7, 2) is perpendicular to the line through (-3, 7) and (2, 2). 5) Find the equation of the line and write in general form a) y-intercept = -3 and x - intercept = 2 b) Through (1, 5) and (2, 6) c) Line parallel to the line x + 3y = 6 through (0, 0) 6) Jim's new car cost $290 per month for car payments. He estimates it costs $.25 a mile for gas and maintenance. a) Express the monthly cost as a function of miles. b) Find the slope of the function. c) What does it cost if he drives 1200 miles this month? 7) Simplify each 8) Solve each by the indicated method. a) (3x -2)(x +4) = 24 by factoring 2 b) 2x - 4x + 5 = 0 by completing the square. 2 c) 7x + x + 1 = 0 by the quadratic formula. 9) Find the vertex point, axis of symmetry, x and y intercepts and graph the function: y 2= x + 2x - 15 210) Rewrite the equation y = 4x + 8x + 2 in the alternate form and state the vertex point. 11) If you drive at x miles per hour and apply your brakes, your stopping distance in 2/25. feet is about f(x) = x + x a) What is your stopping distance at 40 mph? b) What is your stopping distance at 50 mph? c) How much does it increase by from part b to part a? This page hosted by Get your own Free Home Page I am ready to check my answers! Take me there! I need more help! Take me back! Answer Page!! 1) Midpoint is (3, 2) 2) x + y = 5 2x - y = 1 3x = 6 x = 2 Thus the intersection point is (2, 3) 3) a) m = (3 + 3)/(5 - 2) = 6/3 = 2 b) 4x + 5y = 10 5y = -4x + 10 y = (-4/5)x + 2 m = -4/5 c) y = -5 is a horizontal line, thus the slope is zero. 4) Slope for first line is (2 + 3)/(7 - 2) = 5/5 = 1 Slope for the second line is (7 - 2)/(-3 - 2) = 5/(-5) = -1 Thus, they are perpendicular because the slopes are negative reciprocals. 5) a) Use intercept form: x/2 + y /-3 = 1 Multiply by 6. 3x - 2y = 6 b) Use point - slope form. Find slope. It is 1. Use one of the points. y - 5 = x - 1 -x + y - 5 = -1 -x + y = 4 or x - y = -4 c) Find the slope of x + 3y = 6 3y = -x + 6 y = (-1/3)x + 2 The slope is -1/3 Since the line we want is parallel to this line, it has the same slope. (0, 0) is the y-intercept. Use the slope-intercept form. y = (-1/3)x + 0 3y = -x x + 3y = 0 6) a) The fixed cost is 290 and the variable cost is .25m. Thus the equation is C(m) = .25m + 290 b) The slope is .25 c) C(1200) = .25(1200) + 290 = 300 + 290 = 590. $590 7) 8) a) (3x - 2)(x + 4) = 24 2 3x + 10x - 8 = 24 2 3x + 10x - 32 = 0 (x - 2)(3x + 16) = 0 x = 2, x = -16/3 2 b) 2x - 4x = -5 2 x - 2x = -5/2 2 x - 2x + 1 = -3/2 2 (x - 1) = -3/2 x - 1 = c) 9) Vertex point is at -b/2a = -2/2 = -1 Substitute into the equation to find the y value 1 - 2 - 15 = -16 Thus, the vertex point is (-1, -16) Axis of symmetry is the vertical line through the vertex point. x = -1 y-intercept is (0, -15) x-intercepts are found by finding the zeroes of the function 2 x + 2x - 15 = 0 (x + 5)(x - 3) = 0 intercepts are: (-5, 0) and (3, 0) 210) y = 4x + 8x + 2 2 y = 4(x + 2x ) + 2 2 y = 4(x + 2x + 1) +2 - 4 2 y = 4(x + 1) - 2 The vertex point is (-1, -2) 11) a) f(40) = 40 + 1600/25 = 104 ft b) f(50) = 50 + 2500/25 = 150 ft c) 46 ft. This page hosted by Get your own Free Home Page That's it!! Hope you did well on the sample test. I'm all set!! Bring on the real thing!! I'm in deep trouble!! I didn't study near enough!! Take me back to study somemore!!