Algebra 2 with Trig Unit 1 Introduction to ALgebra

Unit 5: Functions
Key Features of Functions

Key
Features of
Functions
Definitions
• Domain: The set of all values of the input
(independent variable x) of a relation or
function
• Range: The range of all values of the output
(dependent variable y) of a relation or function
• X-intercept: The point(s) in which a line
crosses the x-axis (aka the zeros of the
function)
• Y-intercept: The point(s) in which a line
crosses the y-axis
• Average Rate of Change: The average amount
a function changes per unit interval

Domain
• For a function that follows the form of a polynomial, your domain will be
ALL REAL NUMBERS or (-∞, + ∞)
• Examples:
• f(x) = x2
+ 4x + 3
• f(x) = 3x + 9
• f(x) = x + 1
• For functions with fractions with variables on the bottom or radicals with
variables in them, the domain is trickier to get
• Radicals (square roots) can not have negative numbers in them, fractions can not
have a denominator of zero
• Examples:
• f(x) = , f(x) =

Domain (cont.)
• You can also find the domain of a function by looking at its graph
D (-∞, + ∞)
D (-∞, + ∞)

Domain Practice
• What is the domain of the following functions
f(x) = x2
+ 7x + 10
D (-∞, + ∞)
f(x) = |x – 7|
D (-∞, + ∞)
f(x) =
D (-∞,-4) U (-4, + ∞)
f(x) =
D [ -5, + ∞)

Range
• Linear functions are always going to have a range of (- ∞, + ∞)
• Example: f(x) = x + 3
• For quadratics (parabolas), we will want to first find the vertex (lowest or
highest point)
General form for a quadratic: f(x) = ax2
+ bx + c
1. Find the x coordinate of the vertex using:
2. Find the y coordinate of the vertex by plugging in the x-value you just found
3. If a is positive, the range is [y-vertex, + ∞)
If a is negative, the range is (-∞, y-vertex)

Range Practice
• Find the Range of the following functions
f(x) = x2
– 4x + 5
R[1, ∞)
f(x) = -2x2
+ 8x - 5
R(- ∞, 3]
f(x) = x2
– 4x +3
R[-1, ∞]

Range (cont.)
• You can also find the range of a function by looking at its graph
R [0, + ∞)
R [0, + ∞)

X-intercept
• When a line crosses the x-axis (at the x-intercept), the y value is zero
• This means that if we set a function f(x) = 0, we can solve for x to find its x-
intercept(s)
• Examples:
• f(x) = 9 – x
x = 9
• f(x) = x2
+ x – 6
x = -3, 2
• f(x) = 3x + 12
x = -4

Y-intercept
• When a line crosses the y-axis (at the y-intercept), the x-value is zero
• This means that if we set x = 0, we can find the y-intercept
• Examples:
• f(x) = 9 – x
f(0) = 9
• f(x) = x2
+ x – 6
f(0) = -6
• f(x) = 3x + 10
f(0) = 10

Finding X/Y Intercepts by Graphing

Real World Problem
x-intercept: x = 320 mi
y-intercept: y = 16 gal
x-intercept describes the
distance (in miles) the car
has traveled the moment
it runs out of gas
y-intercept describes the
starting amount of gas in
gallons

Positive/Negative Intervals
• A function is positive when f(x) > 0 (above the x-axis)
• A function is negative when f(x) < 0 (below the x-axis)
• We write our answer in interval form using x-values
• Example: In what intervals is this function positive and negative?
Positive: (- ∞, -3) and (3, + ∞)
Negative: (-3, 3)
*Note: we use “(“ and “)” indicating that 3 and -3 are not included, as
they lie on the x-axis and are therefore neither negative nor positive

Increasing/Decreasing Intervals
• A function is increasing on an interval if, for any two points x1 and x2 on that
interval where x1 < x2, we have f(x1) < f(x2)
• All this means is that the function is increasing when y values rise as x
values rise
• Inversely, a function is decreasing on an interval if, for any two points x1 and x2
on that interval where x1 > x2, we have f(x1) > f(x2)
• Again, this means that the function is decreasing when y values fall as x
values fall
• We write our answer in interval form using x-values

Increasing/Decreasing Intervals Practice
• Find the interval where the function is increasing and decreasing
Increasing: (- ∞, 0)
Decreasing: (0, - ∞)
Additional info:
The maximum point of a function is
when it changes from increasing to
decreasing
The minimum point of a function is
when it changes from decreasing to
increasing

Average Rate of Change
What is the average rate of change of a function y = f(x) over the
interval [a,b]?
• We are given our starting point at x = a and an ending at x = b
• The total change in our x values are b - a (final pos. – initial pos.)
• The total change in our y-values are f(b) - f(a) (final pos. – initial
pos.)
• The average rate of change (RoC) will follow the formula:
RoC =
• Does this look familiar?

Average Rate of Change Practice
Find the average rate of change for each graph over the intervals [-2,0] and [0,3]
[-2,0] f(x): RoC = 0 g(x): RoC= 1/2 h(x): RoC = -2
[0,3] f(x): RoC = 0 g(x): RoC= 1/2 h(x): RoC = 3
Bonus: can you tell me what that says about each graph?

Let's put it together
• Find the Domain, Range, Increasing and Decreasing Intervals, and Positive and Negative Intervals
of the following function
f(x) = x2
-6x + 8
Domain: (- ∞, + ∞)
Range: [-1, + ∞)
Increasing: (3, + ∞)
Decreasing: (- ∞, 3)
Positive: (- ∞, 2) U (4, + ∞)
Negative: (2, 4)
• Find the average rate of change over the interval [0,3]
RoC = -3

Algebra 2 with Trig Unit 1 Introduction to ALgebra

Algebra 2 with Trig Unit 1 Introduction to ALgebra

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Algebra 2 with Trig Unit 1 Introduction to ALgebra