Algebra For College And High School Students

Explore one-to-one functions and inverses, use the horizontal line test, and learn step-by-step methods to find inverses with examples of linear, exponential, and rational functions.

Understand the composition and combination of functions by applying sum, difference, product, and quotient rules, and practice open-circle composition (bog and golf) with examples using f(x)=2x+1 and g(x)=x^2+2x-1.

Master the least common denominator by factoring denominators and using the highest powers of each factor. Apply lcd to add or subtract fractions, solve rational equations, and simplify complex fractions.

Solve equations in non rational form by simplifying and collecting like terms, isolating the unknown on one side, and checking for extraneous solutions by substituting into the original equation.

Solve equations in non rational form by foiling, collecting like terms, using order of operations, and verifying solutions across multiple examples.

Learn to solve equations in non rational form by foiling and collecting like terms, then isolate v to obtain 2 and verify for extraneous solutions by substitution.

Solve equations involving fractions by clearing fractions with the lowest common denominator, then collect like terms, isolate the variable, and verify solutions to avoid extraneous results.

Solve equations in fraction form by clearing denominators with the lowest common denominator (lcd), isolate m, and verify results to avoid extraneous solutions.

Solve equations in fraction form using the LCD and factoring, and verify solutions to avoid extraneous results. Apply these steps across examples 4-6 for w, a, and x.

Practice solving equations in fraction form, using the LCD, factoring, and checking for extraneous solutions to confirm x values.

Graph piecewise functions with x ≤ 2 and x > 2, showing y = -2x + 1, (2,-3) dark circle; and y = 5x - 4, (2,6) open circle.

Explore graphing the square root family, including translations, reflections, and stretches, starting from the parent graph y = sqrt(x). Learn to determine domain and range and apply shifts.

explains graphing a square root function with a four left shift, a three downward shift, and a one third vertical shrink. gives domain [-4, infinity) and range [-3, infinity).

Learn to graph polynomials by determining end behavior, y and x intercepts, and multiplicities, using factoring to find zeros, and distinguish even versus odd multiplicities that bounce or cross axis.

Learn to graph polynomials by finding end behavior, y intercept, and zeros with their multiplicities. Identify which zeros cross or bounce and sketch the complete graph.

Graph rational functions by identifying vertical and horizontal or oblique asymptotes, domain, range, and intercepts, with examples of bottom-heavy, equal-degree, and oblique cases.

Master slope formula y2 minus y1 over x2 minus x1, and interpret rise over run to find a line's slope from equation or two points, including vertical and horizontal cases.

Calculate the slope of the line through (5,1) and (2,-4) using the slope formula y2 minus y1 over x2 minus x1, yielding 5/3.

Graph a linear function using rise over run and intercepts, and review plotting points on a Cartesian plane.

Graph the linear line for -2x+y=3 using rise over run method and x- and y-intercepts, learn slope, y-intercept, and converting to y=mx+b for accurate plotting.

Master finding the equation of a linear line in slope-intercept, point-slope, standard, vertical, and horizontal forms using given slopes or coordinates.

Use slope-intercept form to find line through a point with slope, yielding y = -8x - 13; convert y = (1/3)x + 5 to -x + 3y = 15.

Learn to find the equation of a line from two points in slope-intercept, point-slope, and standard forms. Compute slope from (2,5) and (3,9) as 4, then y=4x-3, y-5=4(x-2), and -4x+y=-3.

Identify parallel lines by equal slopes and different intercepts, and perpendicular lines by negative reciprocal slopes; apply these ideas to slope-intercept form problems, including lines through given points.

Explore solving parallel and perpendicular line problems by finding slopes, using negative reciprocals, and writing y = mx + b equations through specified points, including the origin.

Compute the slope from (-3,-2) and (0,0) to get y = (2/3)x, then find a parallel line through (3,1) giving y = (2/3)x minus one.

Solve word problems with linear functions by identifying inputs and outputs, deriving the equation from data, and computing costs using slope and y=mx+b.

Model word problems with linear functions using slope-intercept form. Apply x-y coordinates to heart-rate over time and shipping costs, with S(n) = 0.75 n + 250.

Derive the linear cost equation c(n) for natural gas from the points (1000,39) and (3000,99), obtaining c(n)=3/100 n+9, and find the cost for 2400 cubic feet as 81 dollars.

Practice problems cover slope from coordinates and from equations, with (3,3) and (4,4) giving m=1, and converting standard form to slope intercept form while recognizing horizontal and vertical lines.

Practice problems for linear functions teach writing perpendicular and parallel line equations, converting to standard form, and determining slope and y-intercept, plus basic graphing.

Investigate systems of linear equations in two variables, determining when they are independent, inconsistent, or dependent. Apply substitution, elimination, and graphing, and solve a word problem.

Practice factoring with quiz 1 and determine true or false about the factored polynomial; find x-intercepts by setting f(x) to zero, giving zeros 5, -2, and -7.

Explore simple factoring when a equals one for ax^2+bx+c. Learn to form two parentheses and find two numbers that multiply to c and add to b.

Master simple factoring of quadratics with a=1 and three terms by finding two numbers that multiply to 150 and sum to -25, as in x^2 -25x +150 = (x-15)(x-10).

Learn factoring with the ac method (bottoms up) for quadratics where the leading coefficient isn’t one, rewrite without a, and factor using simple factoring with stepwise examples.

Apply the ac method to factor 4a^2 + 16ab + 15b^2. Split the middle term into 10ab and 6ab, then factor by grouping to obtain (2a+3b)(2a+5b).

learn to factor by grouping for four-term polynomials, split the expression in half, factor out the gcf from each part, and use a repeating factor to verify viability.

Learn to factor and solve polynomials with three terms using the quadratic formula, x = (-b ± sqrt(b^2-4ac))/(2a), especially when the AC method fails.

Solve zeros with the quadratic formula using a = -3, b = 5, c = 6; the discriminant is 97, giving two answers from (-5 ± sqrt(97)) / (-6).

Learn to factor expressions by recognizing the difference of two squares and rewriting them as (a+b)(a-b); apply to examples like 25a^2 - 49b^2 and 144 w^2 - 121 a^2.

Factor the expression nine y squared minus twenty five c squared using the difference of two squares; set a=3y and b=5c to obtain (3y+5c)(3y-5c).

Learn to factor a difference of two cubes using the rule a^3 − b^3 = (a − b)(a^2 + ab + b^2), ensuring two terms, a minus sign, and a power of three, with m^3 − 1 and 8y^3 − 27.

Demonstrates factoring with the difference of two cubes, identifying a and b, and applying (a - b)(a^2 + ab + b^2) to examples such as 125n^3 - 8.

Master the difference of two cubes factoring with x and y, using the rule a cubed minus b cubed equals (a minus b)(a squared plus ab plus b squared).

Practice factoring using the sum of two cubes with a=2t and b=y to get (2t + y)(4t^2 - 2ty + y^2).

learn how to factor out the greatest common factor, identify the GCF of coefficients and variables, and apply factoring rules to factor polynomials completely.

Learn to factor out the greatest common factor across terms, determine the GCF of coefficients and variables, and assess whether the expression factors further.

Factor out the greatest common factor from 12u^5v^6, 18u^2v^3, and -15u^4v^5. Express as 3u^2v^3(4u^3v^3+6-5u^2v^2), and verify no further factoring is possible.

Learn how to factor polynomials using the rational zero theorem and synthetic division, identify possible zeros (x-intercepts) p/q, and determine actual zeros via a remainder check to fully factor polynomials.

Factor polynomials completely using the rational zero theorem and synthetic division, identify p and q, test zeros, and obtain fully factored forms.

Factor polynomials completely using the rational zero theorem and synthetic division, identify a real zero from a candidate like -1/5, and reduce to a quadratic.

Practice factoring techniques from a college algebra lesson, including extracting GCF, factoring two-term polynomials, recognizing perfect squares and differences of squares, and using bottoms-up ac method for quadratics.