Linear Equations in Two Variables & Functions Pre-Calculus 1.2-1.3

SLOPE Slope is defined in three ways… 1) 2) 3) A line with a ______________ slope rises from left to right. A line with a _______________ slope falls from left to right.

FIND THE SLOPE (-3, 2) (5, 12) (8, 3) (-2, -9)

SPECIAL SLOPES A ___________ line has a zero slope A ___________ line has an undefined slope

Parallel & Perpendicular Lines Two lines are ______________ if their slopes are equal Two lines are ______________ if their slopes are opposite reciprocals

Are the two lines parallel or perpendicular? Line A: (-3, 2) (4, -7) Line B: (9, 7) (2, 16) Line A (4, -8) (4, 2) Line B (-3, -8) (8, -8)

SLOPE INTERCEPT FORM y = mx + b y-int slope A line that is written in the linear form, y = mx + b is in slope-intercept form. You can graph these lines by plotting the y- intercept and using the slope to define a second point on the line.

GRAPH! y = ½x – 3 6x – 3y = 12

POINT SLOPE FORM An additional form of a linear equation is called the point-slope form. The equation of a line with a slope, m and passing through (x1, y1) is y – y1 = m(x – x1)

FIND THE SLOPE INTERCEPT FORM OF EACH LINE-- (-1, 3) m = ½ (-5, 9) m = undef. (5, -1) (3, 7)

Find the slope-intercept form of the equation passing through (2, -1) and perpendicular to the line 2x – 3y = 5

A kitchen appliance manufacturing company determines that the total cost in dollars of producing x units of a blender is C = 25x + 3500 Describe the practical significance of the y- intercept and slope of this line.

FUNCTIONS A function is a relation in which each element of the domain is paired with exactly one element of the range… ____________ do not repeat! DOMAIN RANGE

4 WAYS TO REPRESENT A FUNCTION: VERBALLY NUMERICALLY GRAPHICALLY ALGEBRAICALLY

ARE THESE FUNCTIONS?

FUNCTIONAL NOTATION f(x) is read as “f of x” y is a __________________________ can be named ___________________ for f(x) ________________________

f(w) = 4w3 – 5w2 – 7w + 13 f(-2) f(3) f(x)

PIECE-WISE FUNCTION A piece-wise function is defined by 2 or more equations over a specified domain.

𝑦= 𝑥 2 +1, x<0 𝑥−1, 𝑥≥0 GRAPH IT! f(-1) f(0) f(1)

f(x) = 2, 𝑥<−2 𝑥 2 −1, 𝑥≥ −2 Graph it! f(0) f(-2) f(-3)

𝑦= −𝑥 2 +2, x>−1 𝑥+1, 𝑥≤−1 On calculator…. Look at table! f(-2)

FINDING THE DOMAIN OF A FUNCTION The implied domain of a function is the set of all real numbers for which the expression is defined. You must exclude x values that result in division by zero or even roots of negative numbers…. [you cannot divide by zero nor take the square root of negative #]

FIND THE DOMAIN… y= x2 + 3x + 2 f(x) = 5𝑥−8 f(x) = 1 𝑥 2 −4

IN CONCLUSION… *When finding the domain you only have to worry about two special cases!!! Everything else is ALL REAL NUMBERS! Try this one :0) f(t) = 𝑡−1 𝑡−4