Chapter 1: Functions and Their Graphs
1.1 Rectangular Coordinates
A. OBJECTIVE: to plot points in a plane, to calculate distance
and midpoint
B. FACTS/FORMULAS:
1. Cartesian plane = rectangular coordinate system –
axes, origin, quadrants, ordered pair (x, y)
C.
2.
Distance = √(𝑥 − 𝑥)2 + (𝑦 − 𝑦)2
3.
Midpt = (
Ex:
𝑥+𝑥 𝑦+𝑦
2
,
2
)
1.2 Graphs of Equations
A. OBJ: to sketch graphs of equations; to find the intercepts;
to write an equation of a circle and to graph the circle.
B. FACTS:
1. Graph of an equation is the set of all pts. that are
solutions of the equation.
2. Intercepts – pts. where the graph intersects the axes.
a)
(x, 0) and (0, y)
3. Symmetry:
a)
With respect to the x-axis (x,y) → (x, -y)
b)
With respect to the y-axis (x,y) → (-x,y)
c)
With respect to the origin (x,y) → (-x,-y)
4. Circle: (x-h)2 + (y-k)2 = r2 with the Center(h,k) and
radius = r
C. Ex:
1.3 Linear Equations in 2 Variable
A. OBJ: to calculate slope of a line; to write the equation of a
line; to identify parallel and perpendicular lines
B. FACTS/FORMULAS:
1.
2.
3.
4.
C.
Ex:
𝑦−𝑦
Slope = 𝑥−𝑥
a)
Positive
b)
Negative
c)
Zero
d)
No slope
Parallel lines – same slope
Perpendicular lines – opposite/reciprocal slopes
Equation of a line:
a)
Y = mx+b
b)
Y-y1 = m(x-x1)
c)
Ax + By = C
1.4 Fncs
A. OBJ: to identify fncs; to find the domain/range; to
evaluate difference quotients.
B. FACTS/FORMULAS:
1. Domain – input, independent, set of all x values
2. Range – output, dependent, set of all y values
3. Fnc – vertical line test/ x’s cannot repeat
a)
Piecewise fnc- composed of 2+ equations over
a domain
b)
Y=f(x)
4. Difference quotient:
𝑓 (𝑥 + ℎ) − 𝑓(𝑥)
,ℎ ≠ 0
ℎ
C.
Ex:
1.5 Graphs of Fncs.
A. OBJ: to find zeros of a fnc.; to determine where a fnc is
increasing/decreasing; to determine the average rate of
change.
B. FACTS:
1. Fnc.
a)
vertical line test
b)
x values do not repeat
2. Zero of a fnc. – x-intercepts (x,0)
3. Increasing fnc if f(x1) < f(x2) when x1<x2
4. Decreasing fnc if f(x1) > f(x2) when x1<x2
5. Constant fnc if f(x1) = f(x2)
6. Max fnc if f(a) > f(x)
7. Min fnc if f(a) < f(x)
8. Average rate of change = slope
9. Secant line joins any 2 pts. in a fnc.
10. Even fnc – symmetric w/ respect to y-axis f(-x)=f(x)
11. Odd fnc – symmetric w/ respect to the origin f(-x) = f(x)
C.
1.6 Parent Fncs
A. OBJ: to identify and graph a variety of fncs. using the
parent fncs.
B. FACTS/FORMULAS:
1. Parent Fncs:
a)
Linear
b)
Quadratic
c)
Cubic
d)
Square root
e)
Reciprocal
f)
Step
g)
Piece-wise
h)
Greatest Integer
C. Ex:
1.7 Transformations of Fncs.
A. OBJ:
1. Use vertical and horizontal shifts to sketch
graphs of functions.
2. Use reflections to sketch graphs of functions.
3. Use nonrigid transformations to sketch graphs of
functions.
B. Facts/Formulas:
1. Rigid transformations do not change the shape – only
shifts vertically or horizontally
2. Nonrigid transformations distort the shape – shrink or
stretch it.
3. Reflections – flip it over an axis
a)
Over the x-axis: f(x) = - f(x)
b)
Over the y-axis: f(x) = f(-x)
4. Vertical shifts:
a)
f(x) + c is up
b)
f(x) – c is down
5. Horizontal shifts:
a)
f(x+c) is left
b)
f(x-c) is right
1.8 Composition of Fncs
A. OBJ:
1. Add, subtract, multiply and divide fncs
2. Composition of fncs
B. Facts/Formulas:
1. (f+g)(x) = f(x)+g(x)
2. (f-g)(x) = f(x)-g(x)
3. (f*g)(x) = f(x)*g(x)
4. (f/g)(x) = f(x)/g(x) ; g(x)≠0
5. (f o g)(x) = f(g(x))
1.9 Inverse Fncs
A. OBJ:
1. Find inverse functions informally and verify that
two functions are inverse functions of each other.
2. Use graphs of functions to determine whether
functions have inverse functions.
3. Use the Horizontal Line Test to determine whether
functions are one-to-one.
4. Find inverse functions algebraically
B. Facts/Formulas:
1. Inverses – switch x and y.
2. f(f-1(x)) = x or f-1(f(x)) = x
3. The graph of f –1 (x) is a reflection of the
graph of f(x) in the line y = x
4. A fnc. has an inverse if and only if
NO HORIZONTAL line intersects the graph
at more than 1 pt.
1.10 Modeling and Variation
A. OBJ:
1. Use mathematical models to approximate sets of data
points.
2. Use the regression feature of a graphing utility to find
the equation of a least squares regression line.
3. Write mathematical models for direct variation, direct
variation as an nth power, inverse variation,
combined variation, and joint variation.
B. Facts/Formulas:
1. Direct Variation: y = kx
2. Direct Variation as nth power: y = kxn
3. Inverse Variation: y = k/x
4. Combined variation – combines both direct and
inverse variation
5. Joint variation: z = kxy