What is a linear function?

A linear function is a function that can be represented by a straight line on a graph. It has the general form y = mx + b, where m is the slope and b is the y-intercept.

How do you find the slope of a linear function given two points, (x₁, y₁) and (x₂, y₂)?

The slope (m) is calculated using the formula m = (y₂ - y₁) / (x₂ - x₁). This represents the change in y over the change in x between the two points.

What does the y-intercept represent in a linear function?

The y-intercept (b) is the point where the line crosses the y-axis. It represents the value of y when x = 0.

Given the linear function y = 3x + 2, what is the slope and y-intercept?

In the function y = 3x + 2, the slope (m) is 3 and the y-intercept (b) is 2.

If a linear function passes through the points (1, 2) and (3, 6), what is its equation?

First, calculate the slope: m = (6 - 2) / (3 - 1) = 4 / 2 = 2. Using point-slope form with point (1, 2), we get y - 2 = 2(x - 1). Simplifying gives y = 2x. Thus, the equation is y = 2x.

What are the domain and range of a linear function?

The domain of a linear function is all real numbers (R) and the range is also all real numbers (R) as long as the slope is not zero.

How can you graph a linear function using its slope and y-intercept?

To graph a linear function y = mx + b, plot the y-intercept (0, b) and from there use the slope (rise/run) to find another point. Connect the points with a line.

What is the inverse of the linear function f(x) = 3x + 5?

To find the inverse, set y = 3x + 5. Swap x and y to get x = 3y + 5. Solving for y gives y = (x - 5) / 3. Thus, the inverse is f-1(x) = (x - 5) / 3.

If a linear function has a slope of -4, what does this imply about its graph?

A slope of -4 indicates that the line is decreasing, meaning as x increases, y decreases at a rate of 4 units for every 1 unit increase in x.

Which of the following represents a linear function: y = x², y = 2x + 3, y = √x?

The function y = 2x + 3 is a linear function because it is in the form y = mx + b. The others are not linear.

What is the slope of a horizontal line?

The slope of a horizontal line is 0, indicating that there is no change in the y-value as the x-value changes.

If the equation of a line is y = -1/2x + 4, what is the y-intercept?

The y-intercept is 4, which means the line crosses the y-axis at the point (0, 4).

How do you determine if a set of points represents a linear function?

To check if a set of points is linear, calculate the slope between each pair of points. If the slope is constant, the points represent a linear function.

What is the point-slope form of a linear function?

The point-slope form is given as y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.

Given the points (2, 3) and (5, 9), find the slope and write the equation of the line.

The slope is m = (9 - 3) / (5 - 2) = 6 / 3 = 2. Using point-slope form: y - 3 = 2(x - 2). Simplifying gives y = 2x - 1.

What is a piecewise linear function?

A piecewise linear function is defined by different linear expressions over different intervals of its domain.