Week 1 - Lines and Linear Functions

What is the equation for a standard form of a line? What is the equation for a horizontal line? What is the equation for a vertical line? How do you find them? How would you define their slopes?

ax+by=c

• a, b, c are constants

• x and y stay the same

• ex) 9x+16y=72

horizontal line - make a=0, then you get the equation y=k (ex. at y=9, then at the 9 point on the y line of the graph, you make a horizontal line)

• slope = 0/constant

vertical line - make b=0, then you get the equation x=h (ex. x=9, then at 9 on the x line of the graph, you make a vertical line)

• slope is undefined/infinite

What is the formula for the slope of a line?

(y₂-y₁)/(x₂-x₁)

from two points: (x₁, y₁) (x₂,y₂)

When do you know if a linear function is increasing, decreasing, or constant?

slope is positive → increasing

slope is negative → decreasing

slope=0 → constant

Is a line a function if the slope is undefined? Why or why not?

No, because there is only ONE x-value (x=#), and multiple y-values

What is the equation for a point-slope form of a line?

y-y₁=m(x-x₁)

y and x stay the same, x₁ and y₁ are from the first given point, and m is the slope

• use this when finding the equation of a secant line, or in general

• use this if you’re given two points and a slope

What is the equation for a slope-intercept form of a line?

y=mx+k

k is the y-int (0, k), m is the slope, x and y stay the same

What are the requirements for two lines to be parallel?

1. the slopes of the two lines are equal

OR

2. the two lines are both vertical

What are the requirements for two lines to be perpendicular?

1. one line a vertical line, the other is a horizontal line

OR

2. slopes are negative reciprocals

   • m₁ = -1/m₂ and m₂ = -1/m₁

What is the difference quotient? What is the equation for it?

f(a+h)-f(a)/h

the slope of a secant line

Practice problem/concept

x does not equal 2 means the denominator CAN’T be equal to 2