Free Algebra I (Regents) Exam Flashcards

Use inverse operations to isolate the variable: undo addition/subtraction first, then multiplication/division. Whatever you do to one side, do to the other. Example: 3x + 7 = 22 gives 3x = 15, so x = 5.

Solve like an equation, but FLIP the inequality sign whenever you multiply or divide BOTH sides by a negative number. Dividing by a positive number keeps the direction the same.

Treat all other letters as constants and isolate the target variable with inverse operations. For A = lw, divide both sides by l to get w = A/l. The method is identical to solving for a number.

Perform the same operation on all three parts. Subtract 1: -4 < 2x <= 6. Divide by 2: -2 < x <= 3. The solution is all x between -2 (open) and 3 (closed).

Strict (<, >): open circle on a number line, dashed boundary line on a coordinate plane (endpoint NOT included). Inclusive (<=, >=): closed/filled circle, solid line (endpoint included).

Slope m = (y2 - y1) / (x2 - x1), the change in y over the change in x ('rise over run'). Example: through (2,5) and (6,13), m = (13-5)/(6-2) = 8/4 = 2.

y = mx + b. Here m is the slope (rate of change) and b is the y-intercept (value of y when x = 0). It is the easiest form for graphing and for reading the starting value.

y - y1 = m(x - x1). Use it when you know the slope m and any one point (x1, y1) on the line. It lets you write a line's equation without first finding the y-intercept.

Parallel lines have EQUAL slopes (m1 = m2). Perpendicular lines have slopes that are negative reciprocals (m1 * m2 = -1). Example: a line with slope 2 is perpendicular to one with slope -1/2.

Horizontal line: y = c, slope = 0 (no rise). Vertical line: x = c, slope is UNDEFINED (no run, division by zero). A vertical line is not a function.

Slope is the unit rate of change: how much the output changes per 1-unit increase in the input (e.g. dollars per hour, miles per gallon). The y-intercept is the starting/initial value.

Graphing (find the intersection point), Substitution (solve one equation for a variable and plug into the other), and Elimination (add/subtract equations to cancel a variable). The solution makes BOTH equations true.

One solution: lines intersect once (different slopes). No solution: parallel lines, same slope but different intercepts. Infinitely many: the same line (identical equations).

Multiply equations so one variable has opposite coefficients, then add the equations to eliminate it. Solve for the remaining variable, then substitute back to find the other.

Graph both equations; the solution(s) are the point(s) of intersection. A line and a parabola can intersect at 0, 1, or 2 points, so a linear-quadratic system may have up to two solutions.

Graph each inequality (dashed for <, >; solid for <=, >=), shade each solution half-plane, and the OVERLAP of the shaded regions is the solution set. Test a point to confirm which side to shade.

For ax^2 + bx + c = 0 with a not 0: x = (-b ± sqrt(b^2 - 4ac)) / (2a). It solves ANY quadratic equation, even ones that do not factor nicely.

The discriminant is b^2 - 4ac (inside the square root of the quadratic formula). Positive: 2 real roots. Zero: 1 real (double) root. Negative: no real roots (2 imaginary).

The axis of symmetry is the vertical line x = -b/(2a). The parabola is mirror-symmetric across this line, and the vertex lies on it.

Find the x-coordinate with x = -b/(2a), then substitute it back into the equation to get the y-coordinate. The vertex is the minimum point if a > 0 (opens up) or the maximum if a < 0 (opens down).

y = a(x - h)^2 + k, where (h, k) is the vertex. The sign of a tells the opening direction; |a| controls how wide/narrow. Vertex form makes the max/min and axis of symmetry x = h instant to read.

Move the constant aside, take half of the x-coefficient and square it, add it to both sides to form a perfect-square trinomial, write it as (x + p)^2 = q, then take the square root of both sides (keep ±).

If a product equals zero, at least one factor must be zero. So if (x - 3)(x + 5) = 0, then x - 3 = 0 or x + 5 = 0, giving roots x = 3 and x = -5. This is how factoring solves quadratics.

Roots (zeros) are the x-values where the function equals 0. On the graph they are the x-intercepts where the parabola crosses the x-axis. A quadratic has 0, 1, or 2 real roots.

a^2 - b^2 = (a + b)(a - b). Example: x^2 - 25 = (x + 5)(x - 5). It only works for a difference (subtraction) of two perfect squares, never a sum.

Find two numbers that MULTIPLY to c and ADD to b; those numbers become the constants in (x + m)(x + n). Example: x^2 + 7x + 12 = (x + 3)(x + 4) because 3*4 = 12 and 3 + 4 = 7.

Factor out the greatest common factor (GCF) first. Example: 6x^2 + 9x = 3x(2x + 3). Removing the GCF often makes the remaining polynomial easier to factor further.

Multiply First, Outer, Inner, Last terms, then combine like terms. Example: (x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6.

(a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2. Recognizing these speeds up both expanding and factoring. Example: x^2 + 6x + 9 = (x + 3)^2.

Combine like terms (terms with the same variable and exponent). For subtraction, distribute the negative to every term in the second polynomial first. Example: (3x^2 + 2x) - (x^2 - 5x) = 2x^2 + 7x.

Multiplying same base: ADD exponents (x^3 * x^4 = x^7). Dividing same base: SUBTRACT exponents (x^7 / x^2 = x^5). Power of a power: MULTIPLY exponents ((x^3)^2 = x^6).

f(x) names the output rule for input x. To evaluate f(4), substitute 4 for every x and simplify. Example: f(x) = 2x^2 - 3 gives f(4) = 2(16) - 3 = 29.

Domain is the set of all valid INPUTS (x-values). Range is the set of all resulting OUTPUTS (y-values). For a real-world model, the domain is often limited by context (e.g. time cannot be negative).

A graph represents a function only if no vertical line crosses it more than once. This guarantees each input x has exactly one output y. A circle or sideways parabola fails the test.

f(x) + k shifts the graph UP k (or down if k is negative). f(x + h) shifts LEFT h (right if h is negative). a*f(x) stretches vertically; a negative a reflects over the x-axis.

Linear functions change by a constant ADDED amount each step (common difference). Exponential functions change by a constant MULTIPLIED factor each step (common ratio), so they eventually grow much faster.

a is the initial value (y when x = 0) and b is the growth/decay factor. If b > 1 it is growth; if 0 < b < 1 it is decay. A rate r gives growth b = 1 + r and decay b = 1 - r.

It is (f(b) - f(a)) / (b - a), the slope of the line connecting the two points on the graph. For a linear function it is constant; for a curve it varies by interval.

Zeros are inputs where the output is 0 — the x-intercepts of the graph. From an equation, set f(x) = 0 and solve. Zeros are also called roots or solutions.

A sequence with a constant COMMON DIFFERENCE d added to get the next term. Explicit form: a_n = a_1 + (n - 1)d. Arithmetic sequences model linear growth.

A sequence with a constant COMMON RATIO r multiplied to get the next term. Explicit form: a_n = a_1 * r^(n-1). Geometric sequences model exponential growth or decay.

Recursive defines each term using the PREVIOUS term (e.g. a_n = a_(n-1) + 3 with a_1 = 2). Explicit gives any term directly from its position n (e.g. a_n = 2 + 3(n-1)).

r measures the strength and direction of a LINEAR relationship and ranges from -1 to 1. Near +1: strong positive; near -1: strong negative; near 0: little or no linear relationship.

A strong r shows two variables move together, but it does not prove one CAUSES the other. A hidden third factor (a lurking variable) or coincidence can create the correlation.

It is the line that best models a linear trend in a scatter plot. Its slope and y-intercept let you interpret the rate of change in context and predict (interpolate) values within the data range.

A residual is the actual y-value minus the predicted y-value (observed minus model). A residual plot with no clear pattern (random scatter) suggests a linear model is a good fit.

It cross-classifies data by two categories. Joint frequency is a single cell; marginal frequency is a row/column total; relative frequency is a count divided by a total, used to compare proportions or check association.

Four parts, 35 questions: Part I has 24 multiple-choice (2 credits each); Part II has 6 short constructed-response (2 cr); Part III has 4 constructed-response (4 cr); Part IV has 1 constructed-response (6 cr). Max raw score is 82 credits.

A SCALE score of 65 is passing (set by each administration's conversion chart, not a raw 65 percent). A graphing calculator (no symbolic/CAS) and a straightedge are provided, plus a Next Generation Algebra I reference sheet.

Show all work — define the variable, write the equation, and explain reasoning, since partial credit is awarded for correct steps even if the final answer is wrong. Always interpret answers in context.