1. What is a function?
2. What is a linear function?
3. What form does a linear function take? (I.e., What is the standard mathematical notation of a linear function?)
4. What is the formula for determining the slope of a line?
all this is in your text, I am certain. I will be happy to critique your responses.
1. A function is a mathematical concept that describes a relationship between two variables, where each input (or x-value) is associated with exactly one output (or y-value). In simpler terms, a function takes an input and produces a corresponding output based on a predefined rule or formula.
To understand what a function is, you can think of it as a machine that takes an input, performs a specific operation, and gives you an output. For example, a function could be a recipe that takes ingredients (inputs) and produces a dish (output) based on the instructions.
To determine if a given equation or relationship is a function, you can follow this key criterion: for each unique input value, there should be only one corresponding output value. In other words, if you have two different input values that produce the same output, then it is not a function.
2. A linear function is a type of function that represents a straight line on a graph. It is characterized by a constant rate of change between the input and the output variables. In simpler terms, a linear function describes a relationship where the output (or y-value) changes by a constant amount for every unit change in the input (or x-value).
For example, if you have a linear function f(x) = 2x + 3, every time you increase the input x by 1, the output y will increase by 2 units. The slope of the line (which we'll discuss in the next question) determines this rate of change.
Linear functions are often used to model real-world situations where there is a direct proportional relationship between two variables, such as distance and time, or cost and quantity.
3. The standard mathematical notation to represent a linear function is:
y = mx + b
In this notation, y represents the output or dependent variable, x represents the input or independent variable, m represents the slope of the line, and b represents the y-intercept (the point where the line intersects the y-axis).
The slope (m) determines the rate at which the output changes with respect to the input. It represents the ratio of the vertical change (or rise) to the horizontal change (or run) between any two points on the line.
The y-intercept (b) represents the value of y when x is equal to 0, or in other words, it is the point where the line crosses the y-axis.
4. The formula for determining the slope of a line is given by:
m = (y2 - y1) / (x2 - x1)
In this formula, (x1, y1) and (x2, y2) represent the coordinates of two points on the line. By subtracting the y-values and the x-values of these two points and then dividing the result, you can calculate the slope of the line.
The slope describes the steepness of the line: a positive slope indicates an upward slope, a negative slope indicates a downward slope, and a slope of zero indicates a horizontal line. It represents the rate of change between the output and input variables in a linear function.