DOMAIN, RANGE, INTERCEPTS AND GRAPHS OF FUNCTIONS (Constant, Linear, Quadratic, Rational, Piecewise, Absolute Value)

Domain of Constant Function

D = R/ all real numbers

Range of Constant Function

y = f(x) = c
R = c

Graph of Constant Function

horizontal line, y-coord

Domain of Linear Function

D = R / all real numbers

Range of Linear Function

R = R / all real numbers

How to graph a Linear Function

diagonal line.

get x and y intercepts.

How to get x-intercept of Linear Function

set y to 0, and solve for x
(x, 0)

How to get y-intercept of Linear Function

set x to 0, and solve for y
(0,y)

Domain of a quadratic function

D = R / all real numbers

Range of a quadratic function

{y | y >= y-coord of vertex (4ac-b²/4a) } if a > 0
{y | y <= y-coord of vertex (4ac-b²/4a) } if a < 0

How to get vertex of a quadratic function

(-b²/2a , 4ac-b²/4a) = (h , k)

How to graph a quadratic function

parabola shaped. Min/Max point = vertex. Going upwards if a > 0, going downwards if a < 0,

1. Get vertex

2. Get other x values (to the left/right of vertex) and their corresponding y values

3. Copy paste to get left/right values

Domain of a rational function

{ x | g(x) =/ N }

//x should not equal to a N, a value that makes the denominator equal to 0

Range of a rational function

{ y | y =/ N }

//change f(x) to y and solve for x. y should not be equal to N, a value that makes the equation equal to 0

How to get vertical aysmptote

a value that makes the function undefined. ex: x = 1

1. factor out denominator if needed

2. set denominator to = 0

3. solve for x.

values of vertical asymptote are often the same as the domain. there can be multiple vertical asymptotes

How to get horizontal aysmptote

ex: y = 1

d = degree

1. y = 0 if d of numerator < d of denominator

2. y = leading coefficient of numerator / leading coefficient of denominator if d of numerator = d of denominator

3. y = none if d of numerator > d of denominator

Graph of a rational function

hyperbola shaped.

1. Factor out numerator and denominator and determine restrictions

2. Identify asymptotes

3. Identify holes (common factors that cancel out when factoring the numerator and denominator

4. Find x and y intercepts

5. Find additional points (numbers greater than and less than the restrictions)

6. Sketch graph: dashed lines for asymptotes, unfilled circle for holes, etc.

Domain of a piecewise function

values on the right side

Range of a piecewise function

values on the left side

How to graph a piecewise function

1. Evaluate the function at the endpoint/s

2. Plot a few more points to determine shape

3. Repeat for each piece

Domain of an absolute value function

D = R / all real numbers

Range of an absolute value function

R = [0,∞)

//note that this can change if there is a negative sign (-) before the absolute value and/or if there is a constant after (see ppt for more details).

How to graph an absolute value function

v-shaped.

1. general form is: y = a |x - h| + k

2. find vertex by getting (h,k)

3. graph opens upward if a > 0, downard if a < 0

4. graph is wider than y = |x| if |a| < 0

5. graph is narrower than y = |x| if |a| > 1