Honors Algebra 1 Final Review Notes

Solving Linear Equations

• Example: Solve $-3(2x-5)=7-4x$
  • Expand: $-6x+15=7-4x$
  • Add $6x$ to both sides: $15 = 7 + 2x$
  • Subtract 7 from both sides: $8 = 2x$
  • Divide by 2: $x = 4$
• Using Desmos to find the solution: Find the intersection point of the two lines.

Types of Solutions for Systems of Equations

• One Solution: The lines intersect at one point.
• No Solution: The lines are parallel (same slope, different y-intercepts).
• Infinite Solutions: The lines are the same (identical equations).
• Example 1:
  • I. $4-2x=2(2-x) \Rightarrow 4-2x=4-2x$ (Infinite solutions)
  • II. $4x-3=2x-5$ (One solution)
  • III. $8x=5x+3$ (One solution)
  • IV. $12-3x=6-3x$ (No solution)
• Example 2:
  • I. $2(2x-1)+7=2(2x)+5 \Rightarrow 4x + 5 = 4x + 5$ (Infinite solutions)
  • II. $3x+8=2(3x+2)-3x \Rightarrow 3x+8=6x+4-3x \Rightarrow 3x+8=3x+4$ (No solution)
  • III. $8x-2=2(4x-1) \Rightarrow 8x-2=8x-2$ (Infinite solutions)
  • IV. $7(x-3)=2x+3-x \Rightarrow 7x-21 = x + 3$ (One solution)

Domain and Range

• Domain: The set of all possible x-values.
• Range: The set of all possible y-values.
• For a relation given as a set of ordered pairs, list the x-values for the domain and the y-values for the range. Remember not to repeat any values.
  • Example: Given the relation ${(2,2), (3,2), (2,3), (1,4)}$, the domain is ${2, 3, 1}$ and the range is ${2, 3, 4}$.
  • Example: Given the relation ${(-4,5), (-3,4), (0,-1), (2,2), (3,-4)}$, the range is ${5, 4, -1, 2, -4}$.
• For a graphed function:
  • Domain: Read the x-values from left to right.
  • Range: Read the y-values from bottom to top.
  • Use inequality notation to express the domain and range.
  • Closed dots indicate inclusion ($\leq$ or $\geq$), while open dots indicate exclusion ($<$ or $>$).
  • Example: Domain: $-4 \leq x \leq 4$; Range: $-5 \leq y \leq 3$

Solving Formulas

• To solve a formula for a specific variable, isolate that variable using reverse PEMDAS.
• Example: Solve $a(m-r) + t = s$ for $a$.
  • Subtract $t$ from both sides: $a(m-r) = s - t$
  • Divide by $(m-r)$: $a = \frac{s-t}{m-r}$

Linear Functions

• Zero of a function: The x-intercept (where the graph crosses the x-axis).
• Value of the function when x=0: The y-intercept (where the graph crosses the y-axis).

Graphing Linear Equations

• Convert to slope-intercept form ($y = mx + b$), where $m$ is the slope and $b$ is the y-intercept.
• Plot the y-intercept on the y-axis.
• Use the slope to find other points on the line (rise over run).
• Use Desmos to check the work.
• Example: $5x - 3y = 15$
  • $-3y = -5x + 15$
  • $y = \frac{5}{3}x - 5$

Ordered Pairs on a Line

• To check if an ordered pair lies on a line, substitute the x and y values into the equation. If the equation holds true, the point lies on the line.
• Alternatively, put the equation in Desmos and create a table to find if the points are valid.
• Example: Check which ordered pair lies on $3x + 5y = 24$
  • Convert to slope-intercept form: $y = -\frac{3}{5}x + \frac{24}{5}$
  • Points (1, 4.2), (4, 2.4), (5, 1.8), and (8, 0) can be checked. Given the solutions, one can create a table on Desmos with the equation to check.

X and Y Intercepts

• X-intercept: (-3, 0)
• Y-intercept: (0, 3)

Graphing Inequalities

• Solid line: $\leq$ or $\geq$
• Dashed line: $<$ or $>$
• Shade above the line for $>$ or $\geq$
• Shade below the line for $<$ or $\leq$

Systems of Equations

• System of equations: A set of two or more equations with the same variables.
• Example: The length $L$ of a rectangle is 7 less than 3 times the width $W$. The perimeter is 146 meters.
  • $L = 3W - 7$
  • $2(L + W) = 146$
• Substitute 3w-7 for L: $146 = 2((3w-7)+w)$
• Example: Dustin wants to buy 9 tickets to the circus. Lower balcony tickets cost $20.95, and upper balcony tickets cost $12.50. He spent $146.30.
  • Let $x$ be the number of lower balcony tickets and $y$ be the number of upper balcony tickets.
  • $x + y = 9$
  • $20.95x + 12.50y = 146.30$
  • Substituting y = 9-x: $20.95x + 12.50(9-x) = 146.30$

Simplifying Expressions with Exponents

• Use the laws of exponents.
• Example: Simplify $(3 + 1)^2 (3 - 2)^2 (x^2)^2$
  • $4^2 * 1^2 * x^4$
  • $16 * 1 * x^4$
  • $16x^4$
• Another example is $\frac{5x^3y^5}{x^2y^4}$
  • $5 * x^(3-2) * y^(5-4)$
  • $5xy$

Area of a Rectangle

• Area = Length x Width
• Example: The width of a rectangle is $(2x - 3)$, and the length is $(3x + 7)$.
  • $A = (3x + 7)(2x - 3)$
  • $A = 6x^2 - 9x + 14x - 21$
  • $A = 6x^2 + 5x - 21$

Factoring Quadratics

• Example: Factor $6y^2 - y - 12$
  • reverse box method. $(2x-3)(3x+4)$

Lines of Symmetry for Conic Sections

• Vertical line of symmetry: $x = a$

Transformations of Functions

• Vertex form of a quadratic equation: $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.
• To translate a graph left by 2 units, replace $x$ with $(x + 2)$.
• To translate a graph up by 5 units, add 5 to the equation.
• Example: Translate $y = x^2$ left 2 units and up 5 units.
  • $y = (x + 2)^2 + 5$

Solving Quadratic Equations

• Quadratic equations must be set equal to 0 before solving.
• Solve by factoring or using the quadratic formula.
• Example: Solve $3x^2 + 2x + 5 = 6$
  • $3x^2 + 2x - 1 = 0$
  • $(3x - 1)(x + 1) = 0$
  • $3x - 1 = 0$ or $x + 1 = 0$
  • $x = \frac{1}{3}$ or $x = -1$
• Quadratic Formula: For $ax^2 + bx + c = 0$,
  • $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Example: Solve $4x^2 - 3x - 2 = 0$ using the quadratic formula.
  • $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(4)(-2)}}{2(4)}$
  • $x = \frac{3 \pm \sqrt{9 + 32}}{8}$
  • $x = \frac{3 \pm \sqrt{41}}{8}$

Exponential Growth

• Formula: $y = a(1 + r)^x$, where $a$ is the initial amount, $r$ is the interest rate, and $x$ is the number of time periods.
• Example: Betty has $2000 in her savings account, earning 1.5% interest each year. Find the equation to determine the total amount of money $y$ after $x$ years.
  • $y = 2000(1 + 0.015)^x$
  • $y = 2000(1.015)^x$