Algebra I SOL Topics and Formulas
Expressions Follow order of operations PEMDAS Combine like terms (same variables with same exponents) Input Output ( )x∫ = y (output)
X= input (replace every x in the equation with the number in ( )
2(5) 5x x= +∫ Plug in value for x 25 5(5)50
yy= +
=
f(7) = 3x – 5 y= 3(7) – 5 y=16 Equations, Inequalities In calculator Go to Menu Equation Solver Type in the equation exactly how you see it ( include ( ) if needed) Hit exe TWICE!! Inequalities:
Use the same steps as solving equations. **(if multiplying or dividing by a negative FLIP the sign in your answer!!!)
2 147
xx− ≥
≤ −
Function, Patterns Function: x does not repeat (domain is all different numbers) Graphs use vertical line test (can only cross at one point)
EX1: {(3, -4), (3, 5), (4, 6)} EX2: {(7,1), (8,1), (9,1)} EX3 EX4 not a function – the x’s repeat function EX5: EX6: Functions Not a Function – Not a function D:{3,5} R: {4, 10} Not a function
1 2 3
4 5 6
2 5
8 9 10
6 12 1
1 5 5
4 4 10
3 3 5
Dividing by a negative…. flip the sign!
f(x)=abs (x+2)
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-9-8-7-6-5-4-3-2-1
123456789
x
y
x^2/9+y^2/25=1
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-9-8-7-6-5-4-3-2-1
123456789
x
y
x=3
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-9-8-7-6-5-4-3-2-1
123456789
x
y
function not a function – use the vertical line test not a function – use the vertical line test D = { all real numbers} R = { all real numbers greater than 0} Linear Equations, X, Y, Intercepts y=mx+b m=slope b=y-intercept Ax + By = C standard form Horizontal line (HOY) slope = 0 equation starts with y Ex. y=4 Vertical line (VUX) slope = undefined equations starts with x Ex. x=4 To fine the x and y intercept from standard form: Ax + By = C 3x-4y=12 x-intercept y-intercept Cover up the y value Cover up the x value Solve for x Solve for y 3x=12 -4y=12 X=4 y=-3 On Calculator: Solve for y=mx+b Menu Graph Type in equation F6 Draw F5 GSOLV F1 ROOT for the x-intercept F4 YICPT for the y-intercept Slope
Slope= 1 2
1 2
y yx x−
− m=
AB−
Positive Negative Zero Undefined
x
y
x
y
x
y
x
y
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
12345678910
–1–2–3–4–5–6–7–8–9–10
y
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
12345678910
–1–2–3–4–5–6–7–8–9–10
y
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
12345678910
–1–2–3–4–5–6–7–8–9–10
y
Write Linear Equations Y=mx+b slope intercept (where it crosses the y axis)
(m is your slope (riserun
) , b is the y-intercept)
Ax+By=C Standard Form -A, B, and C cannot be fractions
-A cannot be negative - GCF has to be 1
b (y-intercept) Writing an equation of a line from a graph:
1. Find b (y-intercept) 2. Find m (slope) 3. Count up and over to the other point. 𝑟𝑖𝑠𝑒𝑟𝑢𝑛 𝑦 = !!
!𝑥 + 1
4. Plug in m and b into the equation y=mx+b Writing an equation of a line from a point and a slope: (5,-2) m=!!
!
1. Label your point and your slope x=5 y= -2 m=−35 2. y= mx+b plug in the values for y, x, and m y=mx+b
-2=−35 (5) +b 3. Solve for b -2= -3+b
1=b 4. y=mx+b plug in the values for m, and b y=−35 x +1
Writing an equation from two points: (5, -2) (1,0) In Calculator
1. Menu 2. Stats 3. Place x values in L1 4. Place y values in L2 5. F1 Graph 6. F1 Graph 7. F2 x 8. F1 ax+b 9. Replace the a value for a and the b value for b
Systems of Equations: 2 equations, 2 variables, 2 answers
If the 2 lines intersect the solution is (x,y) *you only see two lines touch and make an X*
Slope (m) are different Y-int. (b) are different If they do not cross then they are parallel, no solution *you only see two lines that do NOT touch*
Slope (m) are the same Y-int. (b) are different
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
12345678910
–1–2–3–4–5–6–7–8–9–10
y
Rise
Run
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
12345678910
–1–2–3–4–5–6–7–8–9–10
y
If they are the same line, infinitely many solutions. *you only see one line*
Slope (m) are the same Y-int (b) are the same
On Calculator: Set each formula in standard form Ax By C+ = . Set Each formula in slope intercept form y=mx+b Menu Menu EQUA Graph
F1: Simultaneous Type in both equations F1: 2 unknowns (x and y) F6: Draw Type in A, B, and C for each equation F5: ISCT (they intercept, 1 solution (x,y)) F1: Solve You see 2 lines DO NOT touch, no solution You only see ONE line, infinitely many **If you get MA ERROR you must graph your lines, your answer will be Infinitely many or No Solution. Monomials, Exponents, Scientific Notation Product of Powers-same base Power of a Power
nmnm aaa +=⋅ nmnm aa ⋅=)( Power of a Product Quotient of a Power
mmm baab =)( nmn
m
aaa −=
Power of a Quotient Negative Exponents
x
xx
ba
ba
=⎟⎠
⎞⎜⎝
⎛ a
a
xx 1
=− or
aa x
x=
−
1
0x is always equal to 1 Scientific Notation 2.53 x 510 = 253,000
3.06 x 510− = 0.0000306 Polynomials Add/Subtract like terms
(The same variables with the same exact exponents) 2 43 4 2xy xy x y+ −
2 413xy x y+12 − 2 43 16 15xy xy x y+ −
Multiply/Distribute, FOIL (First Outer Inner Last) ( 5)( 8)x x+ − 2( 2)( 3 8)x x x+ − +
2 8x x− 3 2 8x x x−3 +
5 40x+ − 22x x+ −6 +16
3 2 16x x x− + 2 +
(x+2)(2x-3)
2x2+x-6
Rules for Factoring: 1. Look for a GCF
a. If you find one take it out. In Calculator:
i. Menu ii. Run
iii. Optn iv. F6 v. F4 num
vi. F6 vii. F2 GCD
Type in your #’s with a comma in between. Ex.1 (5, -10, -20)
Ans: 5
Ex 2) 3 2
2
8 568 ( 7)a x aa ax
−
−
Ex 3) 215
5 (3 )a aa a
−5
−1
** When taking out a variable as the GCF, take out the one with the lowest exponent.
Ex: 4 2 2 3 3 2 4
2 2 2 3( )ab x a b x a b xab x b x ab a x
− +
− +
2. If there are 4 terms: a. Find the GCF of ALL 4 terms
4 3 2
3 2
8 64 8(8 64 8)r r r rr r r r
− + +
− + +
b. Break your four terms into two groups
3 2(8 64 )( 8)r r r− +
c. Take out the GCF of the 1st ( ) 3 2
2
(8 64 )8 (r rr r
−
−8)
2 3 40x x− −
d. Take out the GCF of the 2nd ( ) *** you MUST make the ( ) match the first one!!!
( 8)1(rr+
− −8)
** Notice that our ( ) match!!
e. Bring the GCF from step C and D into one set of ( ) and bring one of the matching ones down 2(8 )( 8)r r−1 −
f. If you have a GCF in step a bring that out in front. 2(8 )( 8)r r r−1 −
3. When you have 3 terms. In Calculator:
a. Take out the GCF b. Set the equation equal to zero 2 0Ax Bx C+ + = c. Menu d. EQUA e. F2 Polynomial f. F1: 2nd degree (highest exponent after the GCF) g. Type in the values for A, B, and C (if there is no A, B, or C in your equation type in 0) h. F1: Solve
These are your roots, solutions, zeros to get the factors: place in (x___) with the opposite value.
2 3 4x x− −
[ ][ ]1
4
x
x
− The factors will be (x+1)(x-4)
24x x−5 −6 [ ][ ]3 / 4
2
x
x
− The factors will be (4x+3)(x-2)
***Remember you can check all your work by distributing or doing the box method. Radicals Simplifying Radicals To simplify, use n n nab a b= • or break out into prime factors looking for the same repeated factors (2 or 3 or 4 of a kind—depending on the index).
2
27 3 3 3 3 3of a kind
= =g gEF 354 = 3 3 3 2g g g 33 2=
Comes out of the radical
Simplifying Radicals with variables:
To turn a root into a factor: Take the opposite sign and put in (x __) after x
If you have a fraction as a root: the denominator goes in front of (_x) and the numerator takes the opposite sign and goes after x
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
6
12
18
24
30
–6
–12
–18
–24
–30
y
2325 5 5a pair
x x x x x= =gEF 3 35 7 2 216 2 2 2 2 2 2a b a a a a a b b b b b b b ab a b= =g g g g g g g g g g g g g g g
Solve Quadratic Equation Set trinomial =0 Factor Polynomial like normal (using the calculator) Set each factor (set of parenthesis) equal to zero Solve for the variable You should have two answers!
2 6 27 0( 9)( 3) 0( 9) 09
( 3) 03
x xx xxxxx
− − =
− + =
− =
=
+ =
= −
Vertex: (3,-36)
Domain: R or (-∞,∞) Vertex Range: 𝑦 ≥ −36 (minimum value opens up)
Vertex max opens down
{ }
2
2
120 120 ( 3)( 4)
3, 4
y x xx xx x
x
= − − +
= − − +
= − +
= −
Vertex: (-0.5,12.25) Domain: R or (-∞,∞) Range: 𝑦 ≤ 12.25 Using the calculator: Set equation equal to zero Roots: Y-Intercept:
Menu Menu Graph Graph Type in equation Type in equation F6: Draw F6: Draw F5: GSOLV F5: GSOLV F1: Root F4: Y-ICPT Use the side arrows to go from one root to the other
Vertex: Domain: Always all real numbers or (-∞,∞)
Menu Graph Range: Type in equations Take the y-value from the vertex insert it into equation
Factors
Roots, Zeros, Solutions
Factors
Roots, Zeros, Solutions (Where the graph crosses the x-axis)
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
6
12
18
24
30
–6
–12
–18
–24
–30
y
F6 Draw Graph opens up use 𝑦 ≥ F5 GSOLV Graph opens down use 𝑦 ≤ F2 for MAX (opens down)
F3 for MIN (opens up) Line of Best Fit On calculator: Menu STAT Type all values of X in list 1 Type all values of Y in list 2 F1: GPH1 F1:GPH1 F1:CALC F2: X F1: ax+b Replace a and b with the data given if it is linear
• Look at the scatterplot graph, decide which model is most reasonable (linear) • Calculate the formula using y=ax+b plugging in a, and b
Curve Best Fit On calculator: Menu STAT Type all values of X in list 1 Type all values of Y in list 2 F1: GPH1 F1:GPH1 F1:CALC F2: 2x F6:draw
Look at the scatterplot graph, decide which model is most reasonable (quadratic)
Write Equation using 2y Ax Bx C= + + Plug a, b, and c back into the equation. Round to the second decimal place.
Direct Variation • Equation: y = kx (k is the constant of variation); • graph is a line thru the origin • Solve first equation for k • substitute k into another equation and solve for the unknown variable
Example 1: If y varies directly as x and y is 6 when x is 18, find y when x is 24.
Set up to solve for k: 6 1813
y kxk
k
=
=
=
Then plug k into the formula and find the missing variable
x
y
1 (24)38
y kx
y
y
=
=
=
X Y Direct Variation multiple DIAGONLY across.
II Inverse Variation
• Example – the speed of a car and the time it takes to reach the destination
• Equation: xy k= or kyx
= ( k is the constant variation)
• graph is a hyperbola in opposite quadrants (Quad I & III or Quad II and IV) • To solve find k and substitute it and remaining numbers into eqn. again.
Example 2: If y varies inversely as x and y = 10 when x = 2, find y when x = 6.
If (2)(10)20
xy kk
k
=
=
=
Find y now
6 20206103
xy ky
y
y
=
=
=
=
X Y Inverse you multiply INLINE (across)
2 10
6 y
Statistics Mean- averages (add up all the data and then divide by # of objects) Mode- Most Median- middle # put in order smallest to biggest Range- subtract the lowest value from the highest value
• Standard Deviation, Variance, and Z-score
o Calculator Instructions: Menu Stat F4 Del A (you may need to hit F6 to see Del A on calc) Type in all data into L1 F6 if you don’t see CALC F2 CALC F1 1 VAR σx standard deviation
18 6
24 y
x
y
x̅ is the mean (µ) n is the total number of terms
Variance= (Standard Deviation)2
(std.dev × std.dev) Z-Score
o A positive z-score: above the mean o A negative z-score: below the mean o A z-score of zero: at the mean
o Formula:
Plug in the vaules you have and solve for the unknow Empirical Rule
In a normal distribution with mean µ and standard deviationσ : 68% of the data fall within 1σ of the mean µ 95% of the data fall within 2σ of the mean µ 99.7% of the data fall within 3σ of the mean µ
Creating a Box and Whisker
Calculator Instructions: Menu Stat F4 Del A (you may need to hit F6 to see Del A on calc) Type in all data into L1 F6 if you don’t see CALC F2 CALC F1 1 VAR Scroll down minX is your Lower Extreme Q1 is your lower quartile Med is your Median Q3 is your upper quartile maxX is your Upper Extreme