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

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.

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.

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.

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.

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.

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.

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

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 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.