Digital Forensics

Review for the final exam

               Please review everything that we did so far. The final exam is based on all the chapters that we did .

Let us recall that the definition of the slope of a line is “the change in y over the change in x”. Remember that the parallel lines have equal slopes. For the perpendicular lines, their slopes are negative reciprocals of each other. Let us take some examples.

Examples:

1. Write the slope-intercept form of the line passing through (-1, 5) with x-intercept 4.

Solution: The points are (-1, 5) and (4, 0). The slope = change in y / change in x =  (0 – 5)/ (4 – (-1)) = -5/5 = -1. We have

 y = mx + b, where m = slope = -1. So

y = -x + b. Substitute x = 4 and y = 0.. Then we get b = 4.

So the answer is y = -x + 4.

1. Give the slope and y-intercept of [pic 1].

           Solution: We have to write the given equation in the form y = mx + b.  

            Divide by 2 and write y by itself.                               .

            So y = -2x + 3.. Slope = -2 and the y-intercept = 3.

1. Graph [pic 2].  (Let x =  -2, -1, 0, 1, 2)

Solution:  Substitute the values of x and calculate y. Then we get the following table.

Plot points and we get the graph of [pic 3] .    

[pic 4]                                                       

1.  Use the graph to determine a) the function’s domain, b) the function’s range

     c) x-intercepts, if any d) y-intercepts, if any.

[pic 5]

Solution: (a) Domain: all real numbers or [pic 6].

(b) Range: all real numbers less than or equal to zero.

(c) x-intercept: (1, 0) and y –intercept: (0, -1).

•  Make sure you know how to plot points and graph various functions that we studied.
• To find domains of functions by recognizing categories of functions:

                Category                Domain

                polynomial                all real numbers

                rational                        set denominator [pic 7]

                radical                        set expression under the radical [pic 8]

1. Find the domain and range of the function

                               [pic 9].

         Solution:  It is a polynomial function. We could substitute any number for x and                        calculate [pic 10]. So the domain is all real numbers. To find the range, graph the function and see the y values. The range is all real numbers greater than or equal to 2.

1. Find the domain and range of the function                  

                          [pic 11]

Solution:  It is a rational function. We could substitute any number except 4.

 (since  4 will make the denominator zero). So the domain is all real numbers except 4. The range is all real numbers except zero.

1. Find the domain and range of the function

                               [pic 12].

Solution: The domain is all real numbers greater than or equal to 2. The range is all real numbers greater than or equal to zero.

1. Find the distance between (2, 4) and (5, 8).

Solution: We have to use the distance formula. So the distance is equal to

                [pic 13].

1. Find the midpoint of the line segment with [pic 14] and [pic 15] as endpoints.

      Solution: Using the mid-point formula, the mid-point is

                    [pic 16]

1. Write the standard form of the equation of the circle with center (-8, 5) and  radius[pic 17]

      Solution: The equation is

                  [pic 18]

1. Find the center and radius of the circle whose equation is

                       [pic 19]

      Solution: The center is (2, -5) and the radius is 7.

1. Write the equation of the circle in standard form and give the center and radius.

                [pic 20]

       Solution: We will complete the squares and write the given equation in the standard  form.

                        [pic 21]

So the center is (3, -4) and the radius is 6.

1. Find the coordinates of the vertex for the parabola defined by the given quadratic function.   [pic 22]

             Solution:  Here a = 1, b = 4, and c = 7.

                               [pic 23]     

             So the x-coordinate of the vertex is -2. Find f(-2) to get the y-coordinate of the vertex.  So y = 3. The vertex is (-2, 3).

1. Find the accumulated value of an investment of $8500 if it is invested for 3 years at an

     interest rate of  4.25% and the money is compounded monthly.

Use the compounding interest formula.