Legier Unit 1 review - modeling linear functions

A linear function is a function that can be graphically represented in the Cartesian plane as a straight line. It has the form $$f(x) = mx + b$$ , where $$m$$ is the slope and $$b$$ is the y-intercept.

The y-intercept is the point where the graph of the function crosses the y-axis. It represents the initial value of the function when the input (x) is zero.

The slope (m) can be calculated using the formula $$m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$$ , where (x1, y1) and (x2, y2) are two points on the line.

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

The range of a function is the set of all possible output values (y-values) that the function can produce.

A function is increasing if, as the input values increase, the output values also increase. This is indicated by a positive slope.

A function is decreasing if, as the input values increase, the output values decrease. This is indicated by a negative slope.

A function is linear if the differences between consecutive y-values are constant for equal intervals of x-values.

An example of a linear function is the cost of a gym membership, which includes a fixed registration fee plus a monthly fee.

The slope indicates the rate of change of the function. It shows how much the output (y) changes for a unit change in the input (x).