Algebra 2 Comprehensive Notes

Algebra 2 Assignment Notes

Finding Roots

  • To find all roots of a polynomial equation, set each factor equal to zero and solve for x.

    • Example 1: x(x+5)(x+2)=0x(x + 5)(x + 2) = 0 implies x=0x = 0, x=−5x = -5, or x=−2x = -2

    • Example 2: x(x−2)(x+3)=0x(x - 2)(x + 3) = 0 implies x=0x = 0, x=2x = 2, or x=−3x = -3

    • Example 3: x(3x+4)(x−1)=0x(3x + 4)(x - 1) = 0 implies x=0x = 0, x=−43x = -\frac{4}{3}, or x=1x = 1

Vertex Form, Domain, and Range

  • Rewrite quadratic functions into vertex form: f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex.

  • Completing the Square Method

    • Example 1:

      • f(x)=x2−4x+6f(x) = x^2 - 4x + 6

      • f(x)=(x2−4x+4)+6−4f(x) = (x^2 - 4x + 4) + 6 - 4

      • f(x)=(x−2)2+2f(x) = (x - 2)^2 + 2

    • Vertex: (2,2)(2, 2)

  • Domain: All real numbers (unless otherwise restricted).

  • Range: Determined by the vertex and direction of opening.

  • If a > 0, the parabola opens upwards: [k,∞)[k, \infty)

  • If a < 0, the parabola opens downwards: (−∞,k](-\infty, k]

Transformations of Functions

Vertical Compression

  • Multiply the function by a constant 0 < a < 1.

    • Example: Given y=(x−1)2−4y = (x - 1)^2 - 4, compress vertically to get y=a((x−1)2−4)y = a((x - 1)^2 - 4), where 0<a<1.

Vertical Stretch

  • Multiply the function by a constant a > 1.

    • Example: Given y=(x+3)2−3y = (x + 3)^2 - 3, stretch vertically to get y=a((x+3)2−3)y = a((x + 3)^2 - 3), where a>1.

Translations and Reflections

  • Horizontal Translation: (x−h)(x - h) shifts the graph right by hh units, and (x+h)(x + h) shifts it left by hh units.

  • Vertical Translation: +k+ k shifts the graph up by kk units, and −k- k shifts it down by kk units.

  • Reflection over x-axis: Multiply the entire function by −1-1.

  • Example: Given y=(x−1)2−4y = (x - 1)^2 - 4, translate 3 units left, 5 units down, and reflect over the x-axis:

    • y=−[((x+3)−1)2−4−5]y = -[((x + 3) - 1)^2 - 4 - 5]

    • y=−[(x+2)2−9]y = -[(x + 2)^2 - 9]

    • y=−(x+2)2+9y = -(x + 2)^2 + 9

Locator/Critical Points

Linear Function

  • y=x+4y = x + 4: No critical point, linear function.

  • y=∣x+2∣+4y = |x + 2| + 4: Locator point is (−2,4)(-2, 4). Absolute value function.

  • f(x)=12∣x+1∣+1f(x) = \frac{1}{2}|x + 1| + 1: Locator point is (−1,1)(-1, 1).

  • f(x)=14∣x+2∣+1f(x) = \frac{1}{4}|x + 2| + 1: Locator point is (−2,1)(-2, 1).

Quadratic Function

  • y=(x−3)2−2y = (x - 3)^2 - 2: Vertex (locator point) is (3,−2)(3, -2).

  • y=−2(x+1)2−2y = -2(x + 1)^2 - 2: Vertex (locator point) is (−1,−2)(-1, -2).

Circle

  • (x−3)2+(y−12)2=15(x - 3)^2 + (y - 12)^2 = 15: Center (locator point) is (3,12)(3, 12).

  • (x+5)2+(y−15)2=4(x + 5)^2 + (y - 15)^2 = 4: Center (locator point) is (−5,15)(-5, 15).

Even, Odd, or Neither Functions

Even Function

  • Symmetric with respect to the y-axis.

  • f(−x)=f(x)f(-x) = f(x)

  • Example: f(x)=x4−x2+4f(x) = x^4 - x^2 + 4

Odd Function

  • Symmetric with respect to the origin.

  • f(−x)=−f(x)f(-x) = -f(x)

  • Example: f(x)=xx2−1f(x) = \frac{x}{x^2 - 1}

Neither Function

  • Does not exhibit either even or odd symmetry.

  • Example: f(x)=x+7f(x) = x + 7

Solving Inequalities

Compound Inequalities

  • Solve each inequality separately and find the intersection or union of the solutions.

  • 2x2+5x−3≥x2+7x2x^2 + 5x - 3 \geq x^2 + 7x can be simplified to x2−2x−3≥0x^2 - 2x - 3 \geq 0, which factors to (x−3)(x+1)≥0(x - 3)(x + 1) \geq 0. The solution is x≤−1x \leq -1 or x > 3.

Systems of Equations

  • Determine the number of solutions by graphing or using algebraic methods (substitution, elimination).

  • The number of solutions corresponds to the number of intersection points.

Functions and Inverses

Finding the Inverse

  • Swap xx and yy in the equation and solve for yy.

    • Example: f(x)=2(x−2)5f(x) = 2(x - 2)^5

      • y=2(x−2)5y = 2(x - 2)^5

      • x=2(y−2)5x = 2(y - 2)^5

      • x2=(y−2)5\frac{x}{2} = (y - 2)^5

      • x25=y−2\sqrt[5]{\frac{x}{2}} = y - 2

      • y=2+x25y = 2 + \sqrt[5]{\frac{x}{2}}

Exponential Growth/Decay

Formula

  • A=P(1+r)tA = P(1 + r)^t

    • AA = final amount

    • PP = initial amount (principal)

    • rr = interest rate (as a decimal)

    • tt = time (in years)

  • Example: Initial salary is $51,700 with a 3% raise per year. After 9 years:

    • A=51700(1+0.03)9A = 51700(1 + 0.03)^9

    • A = 51700(1.03)^9 \approx $67,456.77

  • To find the number of years it takes to reach a certain amount, use logarithms.

    • t=ln⁡(AP)ln⁡(1+r)t = \frac{\ln(\frac{A}{P})}{\ln(1 + r)}

    • Example: Current balance is $1,777 with a 4% interest rate. In how many years will the balance reach $2,338.41?

      • t=ln⁡(2338.411777)ln⁡(1.04)≈7t = \frac{\ln(\frac{2338.41}{1777})}{\ln(1.04)} \approx 7

Logarithm Properties

Product Rule

  • log⁡(ab)=log⁡(a)+log⁡(b)\log(ab) = \log(a) + \log(b)

    • Example: log⁡(6x)=log⁡(6)+log⁡(x)\log(6x) = \log(6) + \log(x)

Quotient Rule

  • log⁡(ab)=log⁡(a)−log⁡(b)\log(\frac{a}{b}) = \log(a) - \log(b)

Solving Exponential Equations

Using Logarithms

  • Isolate the exponential term, then take the logarithm of both sides.

    • Example: ex+2=51e^{x + 2} = 51

      • x+2=ln⁡(51)x + 2 = \ln(51)

      • x=ln⁡(51)−2≈3.8918x = \ln(51) - 2 \approx 3.8918

Using Common Bases

  • If possible, rewrite both sides with a common base and equate the exponents.

    • Example: 642p−2=4264^{2p - 2} = 4^2

      • (43)2p−2=42(4^3)^{2p - 2} = 4^2

      • 3(2p−2)=23(2p - 2) = 2

      • 6p−6=26p - 6 = 2

      • 6p=86p = 8

      • p=43p = \frac{4}{3}

  • Another Example:

    • 243−x−3=81−3x−2243^{-x-3} = 81^{-3x-2}

    • (35)−x−3=(34)−3x−2(3^5)^{-x-3} = (3^4)^{-3x-2}

    • 5(−x−3)=4(−3x−2)5(-x-3) = 4(-3x-2)

    • −5x−15=−12x−8-5x-15 = -12x -8

    • 7x=77x = 7

    • x=1x = 1

Law of Sines

  • asin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}

  • Use to find missing sides or angles in a triangle.

Polynomial Graphs

Leading Coefficient

  • Determines the end behavior of the graph.

  • Positive leading coefficient: If the degree is even, both ends point up. If the degree is odd, the left end points down and the right end points up.

  • Negative leading coefficient: If the degree is even, both ends point down. If the degree is odd, the left end points up and the right end points down.

Degree of the Polynomial

  • Determined by the highest power of xx in the polynomial.

  • Even degree: The ends of the graph point in the same direction.

  • Odd degree: The ends of the graph point in opposite directions.

Dividing Polynomials

Long Division

  • Divide as you would with numbers.

  • Example:

    • (x3−11x2+25x+21)÷(x−7)=x2−4x−3(x^3 - 11x^2 + 25x + 21) \div (x - 7) = x^2 - 4x - 3

  • Another Example:

    • (3x4+37x3+95x2+43x−26)÷(x+9)=3x3+10x2+5x−2−8x+9(3x^4 + 37x^3 + 95x^2 + 43x - 26) \div (x + 9) = 3x^3 + 10x^2 + 5x - 2 - \frac{8}{x+9}

Factoring Completely

Common Factors

  • Always look for common factors first.

    • 3x4−17x3+10x2=x2(3x2−17x+10)3x^4 - 17x^3 + 10x^2 = x^2(3x^2 - 17x + 10)

    • =x2(3x−2)(x−5)= x^2(3x - 2)(x - 5)

  • Another Example:

    • −12x2+30x+72=−6(2x2−5x−12)-12x^2 + 30x + 72 = -6(2x^2 -5x - 12)

    • =−6(x−4)(2x+3)= -6(x-4)(2x+3)

Rational Expressions

Restrictions

  • Find values of the variable that make the denominator equal to zero.

    • 1515m−27\frac{15}{15m - 27}, restriction: m≠95m \neq \frac{9}{5}

Simplifying

  • Factor the numerator and denominator, then cancel common factors.

  • Example:

    • r−63r2−18r=r−63r(r−6)=13r\frac{r - 6}{3r^2 - 18r} = \frac{r - 6}{3r(r - 6)} = \frac{1}{3r}

Combining Rational Expressions

  • Find a common denominator, then add or subtract the numerators.

  • Example:

    • 4a+2−53a−3=4a+2−53(a−1)\frac{4}{a + 2} - \frac{5}{3a - 3} = \frac{4}{a + 2} - \frac{5}{3(a - 1)}

    • =4⋅3(a−1)−5(a+2)3(a+2)(a−1)=12a−12−5a−103(a+2)(a−1)=7a−223(a+2)(a−1)= \frac{4 \cdot 3(a - 1) - 5(a + 2)}{3(a + 2)(a - 1)} = \frac{12a - 12 - 5a - 10}{3(a + 2)(a - 1)} = \frac{7a - 22}{3(a + 2)(a - 1)}