Math 1051 Solutions to Review Problems
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1 Math Solutions to Review Problems Homework is to be handed in for grading according to the schedule at the end of the syllabus. Each homework assignment is worth points, which will be awarded as follows: For the tetbook problems, you will earn 3 points if you have done them all and showed how you arrived at the answers. We understand that some problems can be done in your head so no work is needed for those, but most will require some algebra and for those your work must be shown. If you do some but not all of the problems you will be awarded 0,, or points at the discretion of your DIS teacher. The problems at the end of the syllabus are also part of your homework. They are labeled R, where is the corresponding homework number. For eample, R0 is to be handed in with homework HW0. The problems will be graded by your DIS teacher who will award points for a correct solution and answer, point for a competent solution but the wrong answer (e.g., you made a simple arithmetic mistake), and 0 points for a poor solution or a correct answer that is not supported by the proper algebraic steps. Be sure to circle your answers. The solutions will be posted on my web site the day after they are due so you can check your work. Unless specifically asked for, the graphs shown below are not necessary but only serve as checks on the solutions. Tue 6 Sep HW0 is a Diagnostic Pretest. Do homework problems on the key handed out in class. There is no R0. Tue 3 Sep HW0 on Sec A. Geometry. Review problem R0 from Arithmetic. Simplify: 8 First, find LCD : LCD
2 Math Solutions to Review Problems, page HW03 on Sec A.3 Polynomials. Review problem R03 from Sec A. Geometry. Find the area of the small triangle. Let represent length of base of small triangle. Ratios of corresponding sides are equal: A baseheight Tue 0 Sep HW04 on Sec A. Rational Epressions. Review problem R04 from Sec A.3 Polynomials. 3 Factor 30 Factor out GCF first: Now, factor by grouping: a b c 6 ac 6 90 Find two integers with procuct 90 and sum na 8 6 7na 8na 9 Replace with Factor by grouping:
3 Math Solutions to Review Problems, page 3 HW0 on Sec A.6 Equations. Review problem R0 from Sec A. Rational Epressions. Simplify The LCD is so multiply the second term by to make the denominators the same. Then, simplify or
4 Math Solutions to Review Problems, page 4 Fri 30 Sep HW06 on Sec A.8 Word Problems. Review problem R06 from Sec A.6 Equations. 4 Solve 8 u u Let u then u So we have u u 8 u u u or u 9 or or 3 HW07 on Sec A.9 Inequalities. Review problem R07 from Sec A.8 Word Problems. Maren left Minneapolis at :00 pm driving West. An hour later, Sue left from the same place driving East. Maren was driving mph faster than Sue. At 4:30 pm they were 00 miles apart. How fast was each driving? A diagram might help: 00 miles at 4:30 pm Maren Mpls Sue pm pm mph faster Let Maren's rate in mph Then Sue's rate in mph 4 : 30 : hours Maren's drive time 4 : 30 : 00. hours Sue's drive time DistanceMaren DistanceSue 00 miles Rate Maren TimeMaren Rate Sue TimeSue mph Maren's rate 7. mph Sue's rate
5 Math Solutions to Review Problems, page HW08 on Sec A.7 Comple Numbers. Review problem R08 from Sec A.9 Inequalities. Solve 3 7 Get absolute: value by itself on one side: 3 7 Reverse inequality when divide by negative Break up into: two inequalities: or 4 or 6, 6 Using interval notation this is 4, HW09 on Sec A. Radicals. Review problem R09 from Sec A.7 Comple Numbers. Write in comple,, form: i a bi 4 i 4 4 First convert square root of a negative into an imaginary number. Then, multiply numerators and denominators by conjugates of denominators. i i 4 i i 4 4 i i 4 i 4 i 4 i 4 i 4 i 4 i 8 4i 4i i 4 i 6 8i 8i 4i 6 8i 8i 4i 8 8i 4 i i 0 i 0 i i Now, replace i with
6 Math Solutions to Review Problems, page 6 HW on Review Appendi for Eam. Review problem R from Sec A. Radicals. Simplify y Combine the radicals and simplify: y y 0 Since this is a cube root, need TWO additional factors in denominator to make a perfect cube y y y y 4y y y y 4y y y y y y 3 y y 3 4 y y
7 Math Solutions to Review Problems, page 7 Tue 4 Oct HW on Sec. Dist & Midpoint. Review problem R from Sec A.6 Equations. Solve by completing the square: Put the -terms on the same side of the equation and then divide by 3 to make the term have a coefficient of To complete the square, find half the coefficient of the -term and square it or 3 or HW on Sec. Graphing. Review problem R from Sec. Dist & Midpoint. A circle has its center at., 3. The point, is a point on the circle and it is also the endpoint of a diameter. What are the coordinates of the other endpoint of this diameter? The center is at the midpoint of every diameter. So, use the midpoint formula: m. 3 y y ym y 3 6 y y So, the coordinates of the other end point of the diameter are,.
8 Math Solutions to Review Problems, page 8 Tue Oct HW3 on Sec.3 Lines. Review problem R3 from Sec. Graphing. Find the - and y - intercepts of y 3 - intercepts are the values of when y or 3 So - intercepts are the points,0 and,0 3 y - intercepts are the values of y when 0 y y So y - intercept is the point 0, HW4 on Sec.4 Circles. Review problem R4 from Sec.3 Lines. Find the equation of the line that passes through 3, and is perpendicular to the line y 4 Find the slope of y 4 by writing it in y m b form. y 4 y So, m and m Equation of new line is y b Use the given point to find b 3 b 7 b So, the line we want is y 7
9 Math Solutions to Review Problems, page 9 HW on Sec. Functions. Review problem R from Sec.4 Circles. Find the center and radius of the circle y Make the coefficients of the squared terms and then complete the square. y y y 0 To complete the square, find half the coefficient of the -term and square it y y 4 3 y Center:, 0 Radius:
10 Math Solutions to Review Problems, page HW6 on Sec. Graph Functions. Review problem R6 from Sec. Functions. Find domain of f 4 3 Write your answer using interval notation. So far, we have two restrictions on domains:. Cannot divide by 0.. Radicand of a square root must be nonnegative (to avoid imaginary numbers).. Denominators of fractions cannot be zero. First Fraction: or 3 or 4 So,, 4 Second Fraction: 0 0 or So,, Radicand must be nonnegative Now, put them all together., 4,,, Since the inequality says has to be or more, the 4 is taken care of. Therefore, we leave it out. The final answer is,,, Using interval notation we have,,,,
11 Math Solutions to Review Problems, page Fri Oct HW7 on Sec.3 Prop of Functions. Review problem R7 from Sec A.3 Polynomials. 3 Factor 36 3 First, factor out GCF, which is Now, find two whole numbers whose product is ac and whose sum is b 4. They are and 6. Replace the middle term, 4, with 6 and factor by grouping HW8 on Sec.4 Function Library. Review problem R8 from Sec.3 Prop of Functions. This function displays what type of symmetry? f Replace with and simplify. f a a a Recall that b b b f Odd function so symetric about origin HW9 on Sec. Transformations. Review problem R9 from Sec.4 Function Library. if 3 0 Graph f if 6 4 if 6
12 Math Solutions to Review Problems, page Plot points for each part starting at one end of defined interval and ending at other end. Now, check each end point to see if it is included (solid circle) or not (open circle). For y 3, 3 0, 0 included due to 3 not included due to strict inequality 0 For y, 6 not included due to strict inequality 6, 4 not included due to strict inequality 6 For y 4 6, 4 not included due to strict inequality
13 Math Solutions to Review Problems, page 3 HW0 Review of Chapter for Eam. Review problem R0 from Sec.4 Circles. Find the equation of a circle with center, 3 and radius 9. y h y k r or y HW Review of Chapter for Eam. Review problem R from Sec.3 Lines. Find the equation of the line whose -intercept y is,0 and whose -intercept is 0,. Write your answer in slope-intercept form. Slope intercept form is y m b We are given two points so we can find the slope, m. 0 m 0 Now we have: y b We are given that the y-intercept is 0, so b. y
14 Math Solutions to Review Problems, page 4 Tue Oct HW on Sec.6 Math Models. Review problem R from Sec. Transformations. Starting with the graph of f apply transformations that will end with the graph of t 4. As you do each transformation, write the new function and show its graph. g reflects about y -ais f reflects about -ais h j 4 shifts 4 units right k 4 shifts units up
15 Math Solutions to Review Problems, page Tue Nov HW3 on Sec 3. Linear Functions. Review problem R3 from A.3 Polynomials. 3 Divide f by 3. What is the quotient? What is the remainder? Quotient: 6 7 Remainder:
16 Math Solutions to Review Problems, page 6 HW4 on Sec 3.3 Quadratic Functions. Review problem R4 from Sec 3. Linear Funct. Given the following graph, solve f h g algebraically. Write your answer in interval notation. Use the graph to find the equation for each m 6 0 function. For f use 3, 0 and 0, 6 to find slope: y 03 -intercept is 0,6 so 6 is thehorizontalli g is the horizontal line h f h f g h g , 0 ne 6 h X f g
17 Math Solutions to Review Problems, page 7 HW on Sec 3.4 Quadratic Models. Review problem R from Sec 3.3 Quad Functions. Write this quadratic function in verte form: f Complete the square but first factor out 3 to make coefficient of equal to. f To complete the square add and subtract something Find half the coefficient of the -term and square it 3 7 Continued on net page.
18 Math Solutions to Review Problems, page 8 As an alternative, we could write out verte form and then fill in the parts: a h k b 6 ver te a 3 yverte f verte f 3 6 a 7 f The The f 4 7 Now, use any point on the graph to find a. Let's use the y -intercept. y So, when 0, y 4. Now, we can solve for a. 4 a a We actually already knew that a 3 because 3 is the coeffidient of the f term of the original function. So, we have 7
19 Math Solutions to Review Problems, page 9 HW6 on Sec 4. Polynomial Funct. Review problem R6 from Sec 3.3 Quad Functions. Find the equation of a quadratic function that would produce this graph. Write your answer in the form f a b c. It looks like the parabola crosses the -ais at f a 4 4, 0 and, 0 so we know the function has the form To find a, pick another point, say the y-intercept at 0, 8 f a a a So f 4 8 f a OR, start with the verte form f a h k and find values for a, h, and k. The verte is at, 9 so 9 To find a, use another point on the parabola, such as 4, 0. f a a a So, f
20 Math Solutions to Review Problems, page 0 Mon 4 Nov HW7 on Sec 4. Rational Funct. Review problem R7 from Sec 4. Polynomial Funct. 3 Given f 3 a. Find the - and y-intercepts b. State where the graph touches and crosses the -ais. f c. What power function does look like when is very large? d. What function does f look like near where it touches the -ais? e. Sketch the graph of f. Make the horizontal scale and the vertical scale 40 y 0.
21 Math Solutions to Review Problems, page Given f 3 a. -intercepts: Values of when y is 0., 0 3, 0 and 3 so -intercepts are and. y - intercept: Value of y when is 0. 3 y 4 y so -intercept is 0,. b. State where the graph touches and crosses the -ais. Graph crosses when multiplicity of zero is odd. So, cross at 3, 0. Graph touches when multiplicity of zero is even. So, touch at, 0. on does f 3 As, f c. What power function does f look like when is very large? d. What functi where it touches the -ais? Makes f 0 Multiplicity of is even so graph touches -ais when. f look like near g This is a parabola shifted one unit right and stretched by a factor of 8. e. Sketch the graph of f
22 Math Solutions to Review Problems, page HW8 on Sec 4.3 Rational Graphs. Review problem R8 from Sec 4. Rational Funct. 3 3 Given f 9 Find the equations for the vertical and oblique asymptotes f g 3 3 VA when denominator is 0. So, VA is the line 3. 3 The factor 3 cancelled so there is a hole at 3,. To get OA, do division 3 6 So, OA is the line y. 6 HW9 on Sec 4.4 Rational Inequal. Review problem R9 from Sec 4.3 Rational Graphs. 3 3 Graph f 30 Be sure to show intercepts, asymptotes, and holes
23 Math Solutions to Review Problems, page Reduce: f g Since the factors cancelled there is a hole when The y-value at this point is g So the hole is at, 6. Intercepts -i nt is value of when y So -int is, 0. Since 3 3 has odd mult, graph crosses here. 3 y-int is value of y when 0 30 y So y-int is 0, VA This is when denominator is 0, which is when is 6. So, the VA is the line 6 4. HA or OA Since degree of top and bottom are the same we have an HA. It is the ratio of the coefficients of the dominant terms. 3 3 g So the HA is the line y 3 6. Does graph cross HA or OA? That is, for what value of does the function equal the asumptote? g y 3 VA is 3 vertical line is a contradiction so graph does not cross HA. 6. Connect the dots. 7. Plot some points as a check. 3 3 f and f HA is horizontal line y 3
24 Math Solutions to Review Problems, page 4 HW30 Review of Chapter 3 for Eam. Review problem R30 from Sec 4. Poly Functions. Given f 3 What function does f look like where it touches the -ais? Function has zeros when is 0, 3, and. The zero at 3 has multiplicity, which is even, so the graph touches when 3. To find behavior, replace with 3 in the other two factors f f 3 0 Makes f g 6 3 This is a parabola, stretched by a factor of 6 and shifted 3 units right
25 Math Solutions to Review Problems, page HW3 Review of Chapter 4 for Eam. Review problem R3 from Sec 4.3 Rational Graph. 3 Given: f 9 Find the value of where an asymptote of f crosses the graph of f. Degree of numerator is eactly more than degree of denominator so there is an oblique asymptote. Use division to find the equation for the OA: So, OA is the line y To see where it crosses f Function Asymptote f y solve
26 Math Solutions to Review Problems, page 6 Tue Nov HW3 on Sec. Composites. Review problem R3 from Sec 4.4 Rational Inequalities. Solve:. Write inequality with 0 on one side. 0. Find the real zeros of the numerator and denominator Real zeros are at 3,, and Use the real zeros to break up the number line into intervals. Interval: Use a test point in each interval to see if the function is positive or negative. Interval: Try Signs of 3 NN N Sign of f N P N P 0 f? yes no yes no Since we want NP N 0 f, the solution has values of in the intervals from to -3 and also from - to 0. Since the original inequality was not strict (it was rather than ) the -3 and 0 are included. However, we cannot include the - since when = - we would get division by 0, which is not allowed. So, the final, 3,0. interval is NP P PP P
27 Math Solutions to Review Problems, page 7 HW33 on Sec. Inverses. Review problem R33 from Sec. Composites. Given f and g find h f g h and the domain of. h f g f g f For g, For the composition,, can be any real number. So, the domain of h is or,, HW34 on Sec.3 Eponential Functions. Review problem R34 from Sec. Inverses. Find the inverse of f The function is y y y y y y f To find f, echange and y and solve for y. y f Notice how f and are symmetric about the line y. - - f f
28 Math Solutions to Review Problems, page 8 HW3 on Sec.4 Log Functions. Review problem R3 from Sec.3 Eponential Funct. 4 Given 6, find the value of You can morph the given equation into the desired value as follows: 4 6 First, get the 4 out of the eponent: Now, get the out of the eponent: So, is or 6, but that is not possible. 6 Now, convert the to by moving to denominator: 6 Now, get to the numerator: Or, solve the given equation for and then use it in the given epression
29 Math Solutions to Review Problems, page 9 Fri Dec HW36 on Sec. Properties of Logs. Review problem R36 from Sec.4 Log Functions. Solve: 3 log log 0 Note that log log 0 3 log 0 3log log 4 4 HW37 on Sec.6 Log & Ep Equ. Review problem R37 from Sec. Properties of Logs. Write as a single logarithm: log log First, make all the bases the same log 4 log log log 4 log log log 6 log log log log 6 log log log 6 6 log log log 6 log -
30 Math Solutions to Review Problems, page 30 HW38 Review of Chapter for Eam. Review problem R38 from Sec.6 Log & Ep Equ. Solve: 3 Round your answer to one decimal place. ln 3 ln ln3 ln ln 3 ln 3 ln ln ln 3 ln ln ln 3 ln ln 3 ln 3 ln ln 3 ln ln ln Tue 6 Dec HW39 on Sec.7 Models. Review problem R39 from Sec.6 Log and Ep Equations. Solve: log log 4 First, make the bases the same. log log log 4 log log log log 4 log log log 4 log log 4 log
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