Linear Functions
What makes a relation a function?
Each input has exactly one output.
What is another name for rate of change?
Slope
What is the slope-intercept form of a line?
y=mx+b
What makes a sequence arithmetic?
The difference between consecutive terms is constant.
What type of relationship shows when points rise from left to right?
Positive correlation
What makes a function linear?
It has a constant rate of change and graphs as a straight line.
Find the slope between (1, 3) and (4, 9).
(9 – 3)/(4 – 1) = 2
What is the standard form of a line?
Ax + By = C
Find the common difference: 4, 7, 10, 13…
3
What does the slope of a best-fit line represent?
The rate of change between the two variables
Identify the slope and y-intercept in y = 3x – 2.
Slope = 3, y-intercept = –2
What is the general equation for a direct variation?
y=kx
What is the point-slope form of a line?
y – y₁ = m(x – x₁)
Write the explicit formula: 5, 8, 11, 14…
aₙ = 5 + (n – 1)(3)
A scatter plot shows as study hours increase, grades increase. What correlation is this?
Positive linear relationship
Which of the following is not linear? A) y = 4x + 2 B) y = –2x C) y = x² + 1
C) y = x² + 1
Does y = 5x + 7 show a direct variation? Why or why not?
No, because it doesn’t pass through the origin.
Convert y = –2x + 6 to standard form.
2x + y = 6
Find the 10th term of 2, 5, 8, 11…
a₁₀ = 29
A company’s revenue, R, is given by R = 25x + 100, where x is number of products. What does 100 represent?
The starting revenue (y-intercept)
Explain how you can tell from a graph if a function is linear.
The graph is a straight line that passes the vertical line test.
A plumber charges $50 plus $40/hour. What does the slope represent in this linear model?
The $40/hour rate of change in cost per hour worked.
Write the equation of a line with slope –3 through (2, 5) in point-slope form.
y – 5 = –3(x – 2)
How are arithmetic sequences related to linear functions?
Both have a constant rate of change; the common difference acts as the slope.
A factory can produce at most 500 items per day. Write an inequality to represent this constraint.
p ≤ 500