Gearing Up for Honors Geometry!

Transcription

1 Gearing Up for Honors Geoetr! Honors Geoetr is right around the corner and ou need to ake sure ou are read! Man of the concepts ou learned in Algebra I will be used in Geoetr and ou will be epected to reeber the. Please take soe tie this suer and work through this review packet. Refreshing our eor of the concepts learned in Algebra I will help ou hit the ground running in Geoetr in the fall. Even though no one likes to do hoework over suer vacation, putting in a little tie up front will definitel help pa off net ear. This packet is designed to take about a couple hours to do the entire thing so spread out the work. If ou do a little each da, it will be done in no tie! It will be collected net fall b our Honors Geoetr teacher in preparation for our first test. Have a great suer, looking forward to eeting ou in Septeber!! ~Mrs. Vernier (Honors Geoetr teacher at North) Topics Covered in Algebra I that ou need to know for Honors Geoetr Solving Linear Equations Distance Forula Solving Sstes of Equations Midpoint Forula Factoring Pthagorean Theore Solving Quadratic Equations Graphing Lines Siplifing Radicals Writing Equations of Lines Plus ore!! Topics Covered in Middle School that ou need to know for Geoetr Lots of what ou will learn in Geoetr net ear has alread been introduced to ou during our eleentar and iddle school ath eperiences. A basic understanding of shapes, terinolog, and siple area and volue forulas in addition to a good eor of concepts learned in Algebra I will keep ou on the road to success in Geoetr. Thinking Skills Needed for Geoetr Geoetr requires a different tpe of thinking than Algebra I. Algebra I is ostl a procedural course. A step-bstep ethod is taught for how to solve a particular proble and ou get lots of practice astering these skills on siilar probles. In Geoetr, success depends ore on our abilit to think logicall and figure out probles that ou a not have seen an eact replica of before. Spatial relation skills Proble solving skills Logical thinking

2 SOLVING EQUATIONS Solving Linear Equations: SHOW ALL WORK and NO DECIMAL ANSWERS! a. w 5. k 6. 0 w w (80 )

3 z 8 z. 8 Solving Proportions: SHOW ALL WORK and NO DECIMAL ANSWERS! Reeber to solve proportions using cross ultiplication. Here is an EXAMPLE if ou are stuck: OR k 6. a 5 6. k k 5. j j 5

4 ***SOLVING SYSTEMS OF EQUATIONS*** A sste of equations is two equations with two variables, usuall and. You can t solve each equation individuall, but with equations ou can use either substitution or eliination to solve. The solution of the sste is the ORDERED PAIR that works in both equations. If ou reeber that each equation represents a line, ou are just tring to find the ordered pair where the two lines intersect. POSSIBILITIES: One Solution Infinite # of Solutions No Solution Solution The lines intersect and the solution is the ordered pair where the two lines eet. The lines are eactl the sae so lie right on top of each other. The solution is all real nubers or ou can just write the equation of the line as the solution because ever point on the line works. The lines don t intersect because the are parallel so no solution eists. Your answer would be no solution or Ø. Solve using the SUBSTITUTION ethod. Solutions ust be written as an ordered pair, no solution Ø, or the equation of the line! Here is an EXAMPLE if ou are stuck: Since is the sae as, we can replace with in the second equation! 0 Now that ou know, ou can just plug into either equation to find the value of. 0 SOLUTION: (0, )

5 Solve using the ELIMINATION ethod. Solutions ust be written as an ordered pair, no solution Ø, or the equation of the line! Here are two EXAMPLES if ou are stuck: Tr to eliinate one of the variables b adding or subtracting the equations. Soeties the equations are all set for ou like EXAMPLE A, but then there are those like EXAMPLE B when ou will need to force one of the variables to eliinate b ultipling each equation b a nuber. A. 5 6 Now that ou know, ou can just plug into either equation to find the value of. 5 SOLUTION: (, ) B Now that ou know, ou can just plug into either equation to find the value of SOLUTION: (, ) 0.. 6

6 NOW CHOOSE THE METHOD YOU WANT TO USE TO SOLVE THE SYSTEM! ***MULTIPLYING BINOMIALS FOIL*** FOIL- Multipl the FIRST ters, then the OUTER ters, then INNER, and finish with the LAST ters of each grouping. Cobine all like ters for our final answer. Here is an EXAMPLE if ou are stuck:

7 ***FACTORING*** FACTORING- For factoring, ou break up a polnoial into factors or parts, which is the opposite of ultipling like FOIL. There are an tpes of factoring so let s separate the into tpes. GCF-Greatest Coon Factor Pulling out (dividing b) a GCF is the opposite of distributing. Take out the GCF and then write what is left in parentheses. Here is an EXAMPLE if ou are stuck: 9. 6 a a 0. a 8a. 8 6 DOS-Difference of Perfect Squares Here is an EXAMPLE if ou are stuck:

8 Trinoials When ou factor a trinoial, ake sure ou find nubers that ultipl to the last ter and add to the iddle ter. Watch our signs! FOIL our answer to check! Here are two EXAMPLES if ou are stuck: No nuber in front of the squared ter: c 8c c 6c Nuber in front of the squared ter: c c 7. a 6a a a 5. 5k k ***SOLVING QUADRATICS*** Solving Quadratic Equations b Factoring STEPS for solving a quadratic equation, a b c 0, b factoring:. Get the equation equal to zero. Factor. Set each factor or part equal to zero. Solve each equation algebraicall to get the two solutions. The factoring is ied in the practice probles below. Here is an EXAMPLE if ou are stuck:

9 These eaples are alread factored for ou just solve! Factor and Solve! You should have answers for each

10 Solving Quadratic Equations using the Quadratic Forula STEPS for solving a quadratic equation, a b c 0, using the quadratic forula:. Get the equation equal to zero. Deterine a, b, and c. Plug a, b, and c into the forula. Solve algebraicall to get the two solutions. The factoring is ied in the practice probles below. Here is an EXAMPLE if ou are stuck: 5 = 0 b b ac a a = b = 5 c =. After aking sure that one side is equal to 0, identif a, b, and c. Be sure to include the signs on a, b, and c. 5 ( 5). Fill in the blanks of the forula with the values of a, b, c Start under the root and follow the order of operations to siplif and. Once the root is siplified, split the proble into two. 6 6 and 5. If possible, siplif. If not, leave our answer in siplest for. = and =.5 Solve using the quadratic forula. Round all answers to the nearest hundredth

11 ***SIMPIFYING RADICALS NO CALCULATORS THAT DO THE WORK FOR YOU!*** Siplif radical epressions: Break down a radical epression b finding the LARGEST perfect square that divides out of the nuber under the square root. You ight want to review perfect squares: List the largest perfect squares up to and including 5: Here is an EXAMPLE if ou are stuck: Siplif the following radical epressions. NO DECIMAL ANSWERS!

12 Dividing radical epressions: You a not leave a radical epression in the denoinator of a fraction. If ou have a radical in the denoinator, ou will need to RATIONALIZE THE DENOMINATOR. Here is an EXAMPLE if ou are stuck: Siplif the following radical epressions. NO DECIMAL ANSWERS! Distance Forula: Use the distance forula when ou want to find the length of a line segent or the distance between ordered pairs. D Find the distance between each set of ordered pairs. Siplif an radical epressions. NO DECIMAL ANSWERS! 87. (9, 7) and (, ) 88. (0, 6) and (, ) 89. (5, ) and (8, )

13 Midpoint Forula: Use the idpoint forula when ou want to find the idpoint of a line segent with the specified endpoints. Write our answer as an ordered pair. Midpoint, Find the idpoint between each set of ordered pairs. 90. (, ) and (6, 8) 9. (0, 6) and (, ) 9. (, 6) and (8, 6) Pthagorean Theore: Use the Pthagorean Theore to find a issing side of a right triangle. a b c a c b Find the value of. Siplif all radical epressions. NO DECIMAL ANSWERS!

14 ***LINEAR EQUATIONS*** Slope: Slope is the easure of the steepness of a line. To find the slope between two ordered pairs, use the following forula: There are four tpes of slope shown below: Positive Slope Negative Slope Zero Slope Undefined Slope Find the slope between each set of ordered pairs. 96. (, ) and (6, 8) 97. (0, 6) and (7, ) 98. (, 6) and (8, 6) Graphing Linear Equations: For the following linear equations, deterine the slope and -intercept. Then graph the lines on each coordinate plane. See the eaple below to help if ou have forgotten slope-intercept for fro Algebra I: Slope Intercept For : b Equationin SIF : 7 intercept 0, 7 ) Plot the -intercept. ) Use the slope to plot another point on the line. ) Connect the points and draw the line. Put arrows on both ends. up right

15 = = -intercept = -intercept = = = -intercept = -intercept = 0. = -intercept = 5 0. = -intercept = **05. 6 **06. = = -intercept = -intercept =

16 Writing Equations of Lines in Slope-Intercept For: When ou write the equation of a line in slope-intercept for, ou need both the slope and the -intercept (where the line touches the -ais). Soeties ou have this inforation and soeties ou need to do a little work to find the. Reeber that slope-intercept for looks like: b There are a couple ethods ou can use to write an equation in slope intercept for. Use whichever one ou like best because both will give ou the sae equation. Look at the following eaples to choose our favorite ethod: EXAMPLE: Write the equation of the line that goes through the points (, ) and (, ). Use onl slope-intercept for ) Find the slope of the line. 6 ) Find the -intercept b plugging the slope and one of the ordered pairs into slope-intercept for., b b b ) Plug the slope and the -intercept into slope-intercept for and ou are done! Use point-slope for to put the equation into slope-intercept for ) Find the slope of the line. 6 ) Find the equation of the line in slope-intercept for b plugging in the slope and one of the ordered pairs into point-slope for., Write the equation of each line described in slope-intercept for. 07., b 08. and given point, 5 09.

17 0., and,7.,0 and,., 5 and,