Parent Functions Linear Functions f(x) = mx + b Quadratic Functions f(x) = ax +bx + c 2 Cubic Functions f(x) = ax + bx + cx + d 3 Domain: (-∞, ∞)

Range: (-∞, ∞) For Linear Functions Odd function:

f(-x) = -f(x) f(x) = x x-intercept: (-b/m, 0)

y-intercept: (0, b) If m > 0: increasing on (-∞, ∞)

If m < 0: decreasing on (-∞, ∞)

If m=0: constant on (-∞, ∞) Square Root Functions f(x) = √x Step Functions Rational Functions Polynomials of a Higher Order Cube Root Functions For Quadratic Functions f(x) = x 2 For our parent function:

The intercept is at (0,0)

Which is also its minimum

The domain ranges from (-∞, ∞)

The range is from [0, ∞)

The function decreases from (-∞, 0)

And increases from (0, ∞) Remember: For the parent function, m=1 and b=0. For Cubic Functions Domain: (-∞, ∞)

Range: (-∞, ∞)

Intercept: at (0, 0)

Increasing: over interval (-∞, ∞)

Decreasing: never

Odd Function Notice: There are 3 possible x-intercepts for every cubic function. Why is there only one for the parent function? f(x) = √x 3 f(x) = 1/x For Square Root Functions Domain: [0, ∞)

Range: [0, ∞)

Intercept: at (0, 0)

Minimum: at (0, 0)

Increasing: over (0, ∞) Something to think about: Why is the square root function neither odd nor even? 2 f(x) = x 3 f(x) = √x f(x) = int(x) = [[x]] "Greatest Integer Function" Greatest Integer Step Function f(x) = [[x]] Think of it as [[x]] = y, such that y is the greatest integer ≤ x Cube Root Functions Describe the end behavior of both functions. What do you think the degree of each function is? Even Degree Functions What do each of these graphs look like? Red: f(x) = x

Green: f(x) = x

Blue: f(x) = x 2

4

6 For all even parent functions:

Domain: (-∞, ∞)

Range: [0, ∞)

Intercept at: (0, 0)

Minimum at: (0, 0)

Decreasing on interval: (-∞, 0)

Increasing on interval: (0, ∞) How would you describe the end behavior of these functions? Odd Degree Functions Note: In order to find end behaviors of functions, try plugging in x=(-∞) and x=∞ (or essentially really big positive and negative numbers...) For all odd parent functions:

Domain: (-∞, ∞)

Range: (-∞, ∞)

Intercept at: (0, 0)

No minimum or maximum

Increasing on interval: (-∞, ∞) Why do you think that all of these graphs look like the graph of the cubic function? Think about how many x-intercepts you are going to have with each parent function. Red: f(x) = x

Green: f(x) = x

Blue: f(x) = x 3

5

7 How would you describe the end behaviors of any odd-powered parent function? For the cube root parent function:

Domain: (-∞, ∞)

Range: (-∞, ∞)

Intercept at: (0, 0)

Increasing over interval: (-∞, ∞) Can we say that this function is either even or odd? Which one? f(x) = √x 3 Aka as the "floor" of x Domain: (-∞, ∞)

Range: the set of all real integers

x-intercept: over the interval of [0, 1)

y-intercept: (0, 0) Is this function increasing, decreasing, or constant? Think about it for a moment. Rational Functions / The Reciprocal Function *An asymptote is a line or curve that approaches a curve arbitrarily closely. Here, our asymptotic linear equations are:

x = 0

y = 0 Domain: (-∞, 0)U(0, ∞)

Range: (-∞, 0)U(0, ∞)

Decreasing over intervals: (-∞, 0) and (o, ∞) Why do these asymptotes exist? What happens if you plug in a number that is essentially 0 (i.e. x = 1/1000000000...000)?

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# Section 1.6 - Parent Functions

Related to CCSS standard F.IF.7

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