Algebra Module 3-4: Relations & Functions

The value of the function f(x) when x = 7.5.

The transformations applied to the parent function to obtain k(x) include: reflection over the y-axis, a vertical stretch by a factor of 2, a horizontal stretch by a factor of 3, and a translation down 4 units.

The equation for the nth term of the arithmetic sequence 10, 8, 6, 4, ... is an = 10 + (n - 1)(-2).

The slope of the line passing through the points (–3, 5) and (3, 6) is calculated as (6 - 5) / (3 - (-3)) = 1/6.

The value of a function, f(x), at a specific input value, x.

Transformations change the shape or position of a graph.

A vertical stretch multiplies all y-values by a factor greater than 1, making the graph taller.

A horizontal translation shifts the graph left or right.

A vertical translation shifts the graph up or down.

A reflection across the x-axis flips the graph over the x-axis.

The range of a function is the set of all possible output values.

The domain of a function is the set of all possible input values.

The equation of a line given a point (𝑥1, 𝑦1) on the line and the slope 𝑚 of the line.

The equation of a line written in the form Ax + By = C, where A, B, and C are integers and A is positive.

A mathematical equation that describes the relationship between two or more variables. In a linear equation, the highest power of the variables is 1.

The equation of a line written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.

A line that can be drawn to approximate the trend in a set of data points.

A graph that displays the relationship between two variables, usually represented by points.

The measure of how much a line rises or falls for every unit of horizontal change.

The point where a line crosses the y-axis.

The slope of a line represents the rate of change between two points on the line. It indicates how much the y-value changes for every unit change in the x-value.

The point-slope form of a linear equation is a way to represent the equation of a line using the slope (m) and a point (x1, y1) on the line. It is represented as: y - y1 = m(x - x1).

A system of equations is a set of two or more equations that share the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations in the system simultaneously.

To solve a system of equations, we need to find the values of the variables that satisfy all the equations in the system. There are several methods to solve systems of equations, including substitution, elimination, and graphing.

In a real-world problem, we can often set up a system of equations to represent the given information, where the variables represent unknown quantities. By solving the system, we can find the solutions to the problem.

Linear equations are equations that can be written in the form y = mx + c, where m is the slope and c is the y-intercept. They represent a straight line when plotted on a graph.

The y-intercept is the point where the line crosses the y-axis. It is the value of y when x is equal to 0.

The slope of a line is a measure of its steepness. It represents the rate of change in y-values for every unit change in x-values.

A linear inequality that represents a region containing all points that satisfy the inequality. All points on the boundary line are solutions when the inequality is non-strict (≥, ≤).

A line that has a slope of 0; it is horizontal and runs parallel to the x-axis. All points on the line have the same y-coordinate.

A system of equations where one equation’s variable can be expressed in terms of the other variable in the same system. This expression can then be substituted into the other equation to solve for the remaining variable.

A system of equations where the coefficients of one or more variables can be multiplied by constants to create opposite coefficients. The equations are then added together to eliminate one of the variables, allowing you to solve for the remaining variable.

Two lines are parallel if they have the same slope but different y-intercepts. They will never intersect. The slopes will have equal values.

A set of inequalities that represent a solution set that satisfies all the inequalities in the system.

The process of identifying the intersection point of two lines on a graph. At this point, both equations are true.

An absolute value inequality involves the absolute value of a variable or expression. It represents all values that are a certain distance away from a central point.

The slope of a horizontal line is always 0. This is because the y-coordinate remains constant while the x-coordinate changes. Think of it like a flat road - there's no incline, no change in height.

When the x-coordinate is the same for two points, the line is vertical. Vertical lines have undefined slopes. Think of it like trying to find the incline of a wall - impossible because it goes straight up and down.

In the slope-intercept form (y = mx + b), the coefficient of 'x' represents the slope (m), and the constant term represents the y-intercept (b). So, if you have an equation in this form, you can directly identify the slope and y-intercept.

To find the slope of a line given two points, you subtract the y-coordinates and divide by the difference of the x-coordinates. It's simply applying the slope formula: (y2 - y1) / (x2 - x1).

To find the value of 'n' (a missing coordinate) that makes a line pass through two given points with a certain slope, you can plug the known values into the slope formula and solve for 'n'. It's like completing the puzzle with the slope and one point to find the other.