Creating Linear Functions

Front

What is a linear function?

Back

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

Front

What does the slope of a linear function represent?

Back

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

Front

How do you find the slope from two points (x1, y1) and (x2, y2)?

Back

The slope (m) can be calculated using the formula m = (y2 - y1) / (x2 - x1).

Front

What is the y-intercept in a linear function?

Back

The y-intercept is the point where the line crosses the y-axis. It is represented by the value of b in the equation f(x) = mx + b.

Front

How can you determine if a function is linear from a table of values?

Back

A function is linear if the differences between consecutive y-values are constant when the x-values increase by the same amount.

Front

Write the equation of a line with a slope of 2 and a y-intercept of 3.

Back

f(x) = 2x + 3.

Front

What is the value of f(6) if f(x) = 2x + 1?

Back

f(6) = 2(6) + 1 = 12 + 1 = 13.

Front

If the slope of a line is -2, what does this indicate about the line's direction?

Back

A slope of -2 indicates that the line is decreasing; for every 1 unit increase in x, the value of f(x) decreases by 2 units.

Front

What is the value of f(-2) if f(x) = 2x + 1?

Back

f(-2) = 2(-2) + 1 = -4 + 1 = -3.

Front

Write the equation of a line that passes through the points (1, 3) and (3, 7).

Back

First, find the slope: m = (7 - 3) / (3 - 1) = 4 / 2 = 2. Then use point-slope form: f(x) - 3 = 2(x - 1) => f(x) = 2x + 1.