Exponent rules are shortcuts for simplifying powers. When the bases match, multiply by adding the exponents: x^a times x^b = x^(a+b). Divide by subtracting them. For a power of a power, multiply the exponents. For a power of a product or quotient, give the exponent to each factor.
A system of inequalities is two or more inequalities with the same variables. Its solutions are all the points that make every inequality true. To solve it, graph each inequality on the same grid. The solution is the region where the shaded areas overlap.
Systems of equations word problems describe two unknown amounts with two facts. Name a variable for each unknown and write one equation for each fact. Then solve the system by substitution, elimination or graphing. Last, check that the answer makes sense in the story.
An arithmetic sequence is a list of numbers where you add the same amount each time to get the next number. That amount is the common difference, d. For example, 3, 7, 11, 15, … adds 4 each time. The formula a(n) = a(1) + (n – 1)d finds any term.
The average rate of change of a function tells how much the output changes, on average, for each 1 unit of input over an interval. To find it, divide the change in y by the change in x. It is the slope of the line through the two end points of the interval.
Direct variation is a relationship where y = kx for some number k that is not 0. The number k is the constant of variation. When x doubles, y doubles too. The graph is a straight line through the origin. For example, in y = 3x, the constant is 3.
Parallel and perpendicular lines can be spotted by their slopes. Parallel lines have the same slope, so they never meet. Perpendicular lines cross at a right angle, and their slopes are opposite reciprocals. For example, a line with slope 2/3 is perpendicular to a line with slope −3/2.
The standard form of a linear equation is Ax + By = C, where A, B and C are integers and A is not negative. An example is 2x + 3y = 12. The x- and y-terms are on the left, and the number is on the right. This form makes intercepts easy to find.
Point-slope form is a way to write the equation of a line when you know its slope and one point on it. The form is y − y1 = m(x − x1), where m is the slope and (x1, y1) is the point. For slope 3 through (2, 5), the equation is y − 5 = 3(x − 2).
The key features of a function are facts you can read from its graph. They include where it crosses the axes (the intercepts), its highest or lowest point (maximum or minimum), where it goes up or down (increasing or decreasing), and where it is above or below the x-axis (positive or negative).
