Packets  /  Algebra 1 Essentials
Grades 8-10Algebra

Algebra 1 Essentials

The essential Algebra 1 toolkit: functions, linear equations and graphs, systems, polynomials, factoring, and quadratics. Worked examples in the Learn tab, instant-feedback practice, and a 60-second challenge.

1Functions and Function Notation

Key idea: A function is a rule that gives each input exactly ONE output. f(x), read 'f of x', is just a name for y.
  1. Read f(x) as 'f of x' — it is NOT f times x.
  2. To find f(a), replace every x in the rule with a.
  3. Compute carefully, watching the signs.
  4. The result is the output for input a.
Worked example
f(x) = 2x + 1, find f(3)
Replace x with 3: f(3) = 2(3) + 1 = 6 + 1 = 7. Input 3 gives output 7.
Worked example
f(x) = -3x + 5, find f(-2)
f(-2) = -3(-2) + 5 = 6 + 5 = 11. Watch the double negative: -3 times -2 is +6.
Watch out: The vertical line test: if any vertical line hits a graph twice, it is NOT a function — that x would have two outputs!

2Domain and Range

Key idea: The domain is all allowed inputs (x-values); the range is all outputs that actually come out (y-values).
  1. Domain: read LEFT to RIGHT — collect every x that appears.
  2. Range: read BOTTOM to TOP — collect every y that appears.
  3. List each value once (skip repeats).
  4. For graphs, the domain is how far the graph stretches horizontally, the range is how far it stretches vertically.
Worked example
{(0, 1), (2, 3), (4, 5)}
Domain: list the x-values: {0, 2, 4}. Range: list the y-values: {1, 3, 5}.
Worked example
f(x) = x2
You can square any real number, so the domain is all real numbers. Squares are never negative, so the range is y >= 0.
Watch out: Domain is about x (inputs), range is about y (outputs). Mixing them up is the classic mistake!

3Slope and Rate of Change

Key idea: Slope = rise / run = (y2 – y1) / (x2 – x1). Positive slope goes uphill, negative goes downhill, 0 is flat, undefined is vertical.
  1. Label the points (x1, y1) and (x2, y2).
  2. Compute the rise: y2 – y1 (change in y).
  3. Compute the run: x2 – x1 (change in x).
  4. Divide rise by run and simplify the fraction.
Worked example
Slope through (2, 5) and (6, 13)
m = (13 – 5) / (6 – 2) = 8 / 4 = 2. The line rises 2 for every 1 it runs.
Worked example
Slope through (1, 9) and (4, 3)
m = (3 – 9) / (4 – 1) = -6 / 3 = -2. The line falls 2 for every 1 it runs.
Watch out: Keep the ORDER: y2 – y1 over x2 – x1. Subtracting in a different order on top and bottom flips the sign! A vertical line has run = 0, so its slope is undefined.

4Slope-Intercept Form and Graphing Lines

Key idea: y = mx + b is the friendliest line equation: m is the slope, b is where the line crosses the y-axis. Plot b, then use the slope as rise over run.
  1. If the equation is not in y = mx + b form, solve for y first.
  2. Read m (slope) and b (y-intercept).
  3. Plot the y-intercept (0, b).
  4. From there, move by the slope (rise / run) to find a second point, then connect.
Worked example
y = 2x – 5
Compare with y = mx + b: m = 2 (slope), b = -5 (y-intercept). Plot (0, -5), then go up 2, right 1.
Worked example
Rewrite 2x + 5y = 20 in slope-intercept form
Subtract 2x: 5y = -2x + 20. Divide by 5: y = -25x + 4. Slope = -2/5, y-intercept = 4.
Watch out: A horizontal line y = number has slope 0, but a vertical line x = number has UNDEFINED slope — it can never be written as y = mx + b!

5Systems of Equations

Key idea: A system solution is the (x, y) point that makes BOTH equations true. Graph both lines: their crossing point is the answer.
  1. Graphing: draw both lines; the intersection point is the solution.
  2. Substitution: plug the expression for y from one equation into the other.
  3. Solve the one-variable equation for x, then plug back to find y.
  4. Check the point in BOTH original equations.
Worked example
y = x + 1 and y = -x + 5
Set equal: x + 1 = -x + 5. Add x: 2x + 1 = 5. Subtract 1: 2x = 4, so x = 2. Then y = 2 + 1 = 3. Solution: (2, 3).
Worked example
Solve by substitution: y = 3x – 2 and 2x + y = 8
Replace y: 2x + (3x – 2) = 8. Combine: 5x – 2 = 8. Add 2: 5x = 10, so x = 2. Then y = 3(2) – 2 = 4. Solution: (2, 4).
Watch out: Parallel lines never meet: NO solution. Same line twice: INFINITELY many solutions. Always verify your point in both equations!

6Polynomials: Add, Subtract, Multiply

Key idea: Add and subtract polynomials by combining like terms. Multiply two binomials with FOIL: First, Outer, Inner, Last.
  1. Adding: line up like terms and combine the coefficients.
  2. Subtracting: distribute the minus to EVERY term in the second polynomial, then combine.
  3. Multiplying binomials: FOIL — multiply First, Outer, Inner, Last.
  4. Combine the like middle terms and write the final answer.
Worked example
(3x2 + 5x – 2) + (x2 – 3x + 4)
Combine x2 terms: 3x2 + x2 = 4x2. Combine x terms: 5x – 3x = 2x. Constants: -2 + 4 = 2. Result: 4x2 + 2x + 2.
Worked example
(x + 3)(x – 5)
FOIL: x*x = x2, x*(-5) = -5x, 3*x = 3x, 3*(-5) = -15. Combine middles: x2 – 5x + 3x – 15 = x2 – 2x – 15.
Watch out: Subtraction is the danger zone: (5x^2 – 3x + 7) – (2x^2 – 3x + 1) becomes 5x^2 – 3x + 7 – 2x^2 + 3x – 1. The minus flips EVERY sign inside!

7Factoring in a Nutshell

Key idea: Factoring is un-multiplying. Always pull out the GCF first, then check for a quick trinomial pattern. This is a short tour — the Factoring Quadratics module covers every technique in full.
  1. Step 1: factor out the greatest common factor (GCF) if there is one.
  2. Step 2: for x2 + bx + c, find two numbers with product c and sum b.
  3. Step 3: write (x + first)(x + second).
  4. Step 4: check by FOIL — and keep factoring until nothing more works.
Worked example
6x2 + 9x
GCF of 6 and 9 is 3; smallest x-power is x. So 3x(2x + 3). Check: 3x(2x) + 3x(3) = 6x2 + 9x.
Worked example
x2 + 7x + 12
Need two numbers with product 12 and sum 7: 3 and 4. So (x + 3)(x + 4). Check with FOIL.
Watch out: This module only scratches the surface of factoring (GCF and simple trinomials). For difference of squares, perfect squares, the ac-method, and solving by factoring, work through the Factoring Quadratics packet module!

8Quadratic Functions and Graphing

Key idea: y = ax^2 + bx + c graphs as a parabola. If a > 0 it opens up (vertex is the minimum); if a < 0 it opens down (vertex is the maximum). Vertex x = -b / (2a).
  1. Identify a, b, c in y = ax2 + bx + c.
  2. Find the vertex x-coordinate: x = -b / (2a).
  3. Plug x back in to get the vertex y-coordinate.
  4. Plot the vertex, find the x-intercepts (zeros) by factoring if possible, and sketch the curve.
Worked example
y = x2 – 4x + 3: find the vertex and zeros
Vertex x = -b / (2a) = 4 / 2 = 2. Plug in: y = 4 – 8 + 3 = -1. Vertex: (2, -1). Factor: (x – 1)(x – 3) = 0, so the graph crosses the x-axis at x = 1 and x = 3.
Worked example
y = -2x2 + 8x – 6: which way does it open?
a = -2, which is negative, so the parabola opens DOWNWARD — the vertex is the highest point.
Watch out: The vertex is a MINIMUM when a > 0 and a MAXIMUM when a < 0. Do not mix them up! The parabola is symmetric: the zeros sit the same distance left and right of the vertex x-value.

9Exponent Rules Review

Key idea: Same base: multiply = add exponents, divide = subtract exponents. Power of a power = multiply exponents. Different bases = leave them alone.
  1. Check the bases match — no matching base, no rule.
  2. Multiplying powers: add the exponents.
  3. Dividing powers: subtract the exponents.
  4. Power of a power: multiply the exponents. Power of a product: distribute the power to every factor.
Worked example
x3 * x5
Same base: ADD exponents. x3 * x5 = x^(3+5) = x8.
Worked example
(2x4)2
Power of a product: raise EACH part. 22 = 4 and (x4)2 = x8. Result: 4x8.
Watch out: You can only combine exponents when the BASES are the same: x^3 * y^5 cannot simplify! And (x^3)^2 = x^6, not x^9 — power of a power MULTIPLIES the exponents.
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60-Second Challenge

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