Precalculus Graphing

Domain
1. If There's a Denominator:
  1. Determine all values of x for which the denominator is = 0.
    These values WILL NOT be in the domain.
  2. If the denominator involves a radical, determine all values for which the radicand is positive:
    1. Factor the radicand and find its zeros. Plot these on a number line.
    2. For each section of the number line in the previous step, select one number (not the zeros you plotted!) and determine if the radicand is positive or negative for that value.
    3. The denominator's contribution to the domain of the function will exclude the zeros found above along with any region of the number line that renders any radicand in the denominator negative.
2. Examine the Numerator:
  1. If the numerator involves a radical, determine all areas of the number line for which the radicand is < 0, using the method mentioned in section 2 a above.
  2. The numerator's contribution to the domain of the function will exclude any regions found in the above step.
3. Combine the Domains:
  1. On two separate, parallel number lines, trace out the domains found in sections A 2 and B 2 above.
  2. The DOMAIN OF THE FUNCTION will consist of those regions / values on the number line that are COMMON to the two number lines traced in the previous step.
  3. NOTE: If there is no common area, then the function does not have a domain, and there IS NO GRAPH!
Zeros
1. Express the function in simplest (reduced) form.
2. The zeros will be any values of x IN THE DOMAIN DETERMINED IN SECTION I that render the numerator equal to 0.
3. Note: If the numerator has zeros that are NOT in the (combined) domain of the function, then they ARE NOT zeros of the function.
Vertical Asymptote(s)
1. If the function does not have a denominator, it will NOT have any vertical asymptotes.
2. The vertical asymptotes occur at every zero of the denominator that is included in the "combined" domain and which DOES NOT FACTOR & REDUCE with a corresponding factor in the numerator.
3. The format of a vertical asymptote's equation is: x = value.
4. If a zero of the denominator is included in the "combined" domain and DOES "cancel":
  1. There will NOT be an asymptote at this value -- there will simply be a "hole" in the graph.
  2. To find the y-coordinate of this hole, substitute the x value into the reduced form of the function found in section II A above.
Horizontal / Other Asymptote(s)
1. If there is no denominator, there will not be a horizontal / other type of asymptote.
2. If the domain is not infinite, there will not be a horizontal / other type of asymptote.
3. If there IS a denominator and the function has an infinite domain:
  1. Compute the limit of the function as x approaches infinity:
    1. Compare the degrees of the numerator and denominator.
    2. If the degree of the numerator is lower, the limit is 0 and there is a horizontal asymptote at y = 0.
    3. If the degree of the numerator is equal to the degree of the denominator, the limit is = the ratio of the leading coefficients, and there is a horizontal asymptote at y = ratio.
      - Beware: If your function is basically [SQRT(x²)]/x, then you will have two horizontal asymptotes:
        y = +1 (For positive values of x) and y = -1 (For negative values of x).
    4. If the degree of the numerator is higher, you must divide the denominator into the numerator. The asymptote will be
      y = (quotient) [Do NOT include the "remainder"]
Intersection with Asymptote(s)
1. Set the formula for your function) equal to the "ratio" or "quotient" found in sections c and d immediately above.
2. Solve for x. Eliminate any values not in domain.
Symmetry
1. Sketch the graph, using all the information you have gathered so far.
2. Keep in mind:
  1. The function can only change from positive (i.e. above the x-axis) to negative (below the x-axis) at values where:
    1. The function has a zero
    2. The function has a vertical asymptote
    3. The is some other break in the domain
  2. Observe the graph to make conclusions about symmetry.
  3. For your reference:
    1. f(x) is EVEN if f(-x) = f(x) for all x in the domain.
      - Even functions are symmetric with respect to the y-axis.
    2. f(x) is ODD if f(-x) = - f(x) for all x in the domain.
      - Odd functions are symmetric with respect to the origin.
    3. Function can be symmetric with respect to unexpected lines, such as x = 2.

Domain: x > 3, x < - 3.
Zeros: x = 3, - 3
VA: none
HA: y = 1, y = - 1
Intersection: (5, 1)
Description of Graph:

Domain: - 2 < x < 0, x >1.
Zeros: x = - 2, 0, 1
VA: none
HA: none
Description of Graph:

Domain: x does not equal 5
Zeros: none
VA: x = 5
HA: y = 0
Description of Graph: