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# Linear Functions

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## Nicholas Redding

on 19 February 2013

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#### Transcript of Linear Functions

The Answers Positive and Negative Slopes Linear Functions Slope (m): m=(y2-y1)/(x2-x1) or
m=rise/run Transformations Horizontal and Vertical Lines Horizontal Lines Word Problem Slope Intercept Form: y=mx+b
Point Slope Form: y-y1=m(x-x1)
Standard Form: Ax+By=C Domain: (-∞,∞)
Range: (-∞,∞) Slope is always constant. Positive Slope Parent Function: f(x)=x or
Equation: y=x Zeros: x=(0,0)
Y-intercept: y=(0,0) Minimum: n/a
Maximum: n/a Even, Odd, or Neither? Odd! End Behavior: As x -> ∞, f(x) -> ∞
As x -> -∞, f(x) -> -∞ Increasing=(-∞,∞)
Decreasing=n/a Negative Slope Slope is always constant. End Behavior: As x -> ∞, f(x) -> -∞
As x -> -∞, f(x) -> ∞ Increasing=n/a
Decreasing=(-∞,∞) An example of a Positive Slope Equation
goes as follows: y=2x+5.
The 2 in front is positive, so the slope is
positive. An example of a Negative Slope Equation
goes as follows: y=-2x+5. (as shown in the graph above.)
The 2 in front is negative, so the slope is
negative. Negative Slope is has almost the
exact same properties of Positive
Slope with a few exceptions. Slope (m): m=(y2-y1)/(x2-x1) or
m=rise/run Slope Intercept Form: y=mx+b
Point Slope Form: y-y1=m(x-x1)
Standard Form: Ax+By=C Parent Function: f(x)=-x or
Equation: y=-x Domain: (-∞,∞)
Range: (-∞,∞) Zeros: x=(0,0)
Y-intercept: y=(0,0) Minimum: n/a
Maximum: n/a Even, Odd, or Neither? Odd! Parallel and Perpendicular Lines Parallel Lines Parallel Lines have an equal slope and will never intersect.
They have the same properties as a Linear Function. Perpendicular Lines Perpendicular Lines have opposite reciprocal slopes and will intersect at 90° angles. They have the same properties of a Linear Function An example of equations with parallel lines are as follows: If you solve for y:
-2x+y=9
+2x +2x
y=2x+9 If you solve for y:
6x-3y=5
-6x -6x
-3y=-6x+5
/-3 /-3
y=2x-(5/3) Since both slopes (y=mx+b) are 2x, this creates Parallel Lines. An example of equations with Perpendicular Lines are as follows: To create a Perpendicular Line, you must have the opposite reciprocal slope. y=2x+3
to
y=(1/2)x-1 The slopes are now opposite reciprocals.
(2/1) to (1/2) Vertical Lines Horizontal Lines are a special type of Linear Graph. Slope (m): m=0 Y-intercept: y=constant Zeros: This is a special case and only happens when y=0. When this happens, x=∞. Domain: (-∞,∞)
Range: {constant} An example Horizontal Line equation would look as follows: y=-2.3. (as shown in the graph above.) Vertical Lines are another special type of Linear Graph. Slope: undefined Domain: {constant}
Range: (-∞,∞) Zeros: x=constant Y-intercept: This is a special case and only happens when x=0. When this happens, y=∞ An example Vertical Line equation would look as follows: x=1.4. (as shown in the graph above.) y=mx-b, where subtracting b to a function moves the graph down b positions. y=b(mx), where multiplying a function by b stretches the graph vertically b times (lengthens the graph by a scale factor of b). y=mx+b, where adding b to a function moves the graph up b positions. y=1/b(mx), where dividing the function by b compresses the graph vertically (shortens the graph by a factor of b, makes it 1/b times as tall). y=-mx, where multiplying a function by -1 reflects the graph over the x-axis (creates a mirror image of the graph across the horizontal x-axis). Jack and Jill are going on vacation to Las Vegas, Nevada for one week. Jack looks on Travelocity for plane tickets and finds them for \$637 per person (round trip). He finds a hotel room for \$549 per person (per night). He figures that they will need to add around \$800 for meals and other activities. Jill looks on Expedia for plane tickets and finds them for \$412 per person(one way). She finds a hotel room for \$1021 (per night) that covers all people staying in the room. She figures that they will need to add around \$900 for meals and other activities. Equations:
Jack--y=637p+549pn+800
Jill--y=412fp+1021n+900 Let p=persons
n=nights
f=flights 1. Jack--y=637(2)+549(2)(6)+800
y=1274+6588+800
y=\$8662

Jill--y=412(2)(2)+1021(6)+900
y=1648+6126+900
y=\$8674

Jack found the better deal. 2. Jack--y=637(3)+549(3)(6)+800
y=1911+9882+800
y=\$12593

Jill--y=412(2)(3)+1021(6)+900
y=2472+6126+900
y=\$9498

Jill now has the better deal. 1. Which person has found the better deal?
2. What if they decide to bring someone else with them?
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