16
Remark. To find x- and y-intercepts algebraically:
• Set x = 0 for y-intercept
Note: For linear equations in the form y = mx + b, the y-intercept is just (0, b)
• Set y = 0 for x-intercept(s)
Example. Find the x- and y-intercepts for the graph of each equation.
2
y = 2x + 7
y=
x+1
3
x-intercept(s)
x-intercept(s)
y-intercept
y-intercept
x=
3
y=5
x-intercept(s)
x-intercept(s)
y-intercept
y-intercept
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Applications of Linear Functions–Average Rate of Change (Section 1.3)
Remark. The average rate of change between any two data points on a graph is the slope of
the line passing through the two points.
Example. Kevin’s savings account balance changed from $1140 in January to $1450 in April.
Find the average rate of change per month.
Example. A plane left Chicago at 8:00am. At 1:00pm, the plane landed in Los Angeles, which
is 1500 miles away. What was the average speed of the plane for the trip?
Example. Find the average rate of change for each function on the given interval.
f (x) = x3 + 2x2 x on [ 1, 2]
f (x) = 7x 3 on [4, 7]
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Applications of Linear Functions–Modeling (Section 1.4-1.5)
Example. Given the following information, determine a formula to convert degrees Celsius to
degrees Fahrenheit.
• Water freezes at 0 Celsius and 32 Fahrenheit.
• Water boils at 100 Celsius and 212 Fahrenheit.
Example. A taxi service charges a $3.25 pickup fee and an additional $1.75 per mile. If the
cab fare was $17.60, how many miles was the cab ride?
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Parallel and Perpendicular Lines (Section 1.4)
Definition. Two distinct lines are called parallel i↵
(1) they are both vertical lines, or
(2) they have the same slope
Example. Determine if the two lines are parallel.
6x
4y
2y = 2
8 = 12x
4y 7 = 8x
2y + 4x = 1
Example. Find the equation of the line through the given point and parallel to the given line.
(0, 3), y =
x + 11
(9, 1), y =
4
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1
(2, 5), y = x + 3
2
(5, 4), x = 2
Definition. Two distinct lines are called perpendicular i↵
(1) one is vertical and one is horizontal, or
(2) their two slopes, m1 and m2 , satisfy m1 · m2 =
1
Example. Determine if the two lines are perpendicular.
y = 2x + 4
4y + 2x + 1 = 0
5y = x
5x y =
4
21
Example. Find the equation of the line through the given point and perpendicular to the given
line.
(0, 3), y =
x + 11
1
(2, 5), y = x + 3
2
(9, 1), y =
(5, 4), x = 2
4
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Intersection Points and Zeros (Section 1.5)
Definition. An intersection point of two functions is a point (x, y) where the graphs of each
function cross (i.e. it is a point that is in common to both graphs).
Example. Find the intersection point (if any) between the two given lines.
y = 3x + 2 and y = 5x 6
y = 2x 5 and y = 2x + 8
y=
3
4
x + 2 and y = x
4
3
5
y = 5 and x =
2
Example. A private plane leaves Midway Airport and flies due east at a speed of 180 km/h.
Two hours later, a jet leaves Midway and flies due east at a speed of 900 km/h. After how many
hours will the jet overtake the private plane?
23
Definition. Zeros of a function are the values of x that make the function equal 0 (also the
x-coordinate of the x-intercept).
Example. Find the zero(s) of each function (if any).
f (x) = 8x + 4
g(x) =
1
g(x) = x
2
f (x) = 4x + 3
3
5
Example. A plane is descending from a height of 28,000 feet at a constant rate of 300 feet per
minute. After how long will the plane have landed on the ground?