Maximums, Minimums, and Everything Related

Reading Graphs

  1. The graphs will usually look like squiggly lines containing parabolas. You are looking to label a number of things on these graphs.

Domain

  1. Domain is the range of X values in the graph

  2. Mark the Domain using intervals

  3. an interval is a type of notation where you are not noting (x, y) but more (beginning value, end value). When using intervals, Use [brackets] to indicate firm position. Use (parentheses) to indicate open position. Parentheses are always used with infinity signs.

  4. Find the Domain by finding the first x value farthest to the left, now find the farthest to the right. A domain might look like (-5, 8] indicating not including but open -5 and firm 8. Note that the first value must be smaller than the second value or its not an interval

Range

  1. Range is the range of Y values in the graph

  2. Mark the Range using intervals

  3. Find the Range by finding the y value that is the farthest down, next find the one farthest up. Example: [-1, 3]

Increase

  1. Increase is where the lines go up

  2. To mark the increase, mark the X VALUES of the places the line starts going up and stops going up

  3. Increase should be marked as an interval. If there is more than one increase, it should be marked with a Union, a union is represented by U and shows the two sets are related. It is like saying the line increases here AND (U) here.

  4. Example: (-3, -1)U(2, 4). note the x values must be in INCREASING ORDER. You cannot have (4, -1)

  5. Mark these with parentheses

Decrease

  1. Decrease is where the lines go down

  2. To mark the Decrease, mark the X VALUES of the places the line starts going down and stops going down

  3. Decrease should be marked as an interval

  4. Example: (-5, -3)U(-1, 2)U(4, 8). Note that the x values are still in increasing order because we are marking the decreases going left to right so the x can only get bigger

  5. Mark these with parentheses

Constant

  1. Constant is where the line stays flat

  2. To mark the Constant, mark the beginning and ending x value of the flat line

  3. Constant should be marked as an interval

  4. Example: (-5, -3)

X Intercept

  1. The X intercept is anywhere the line crosses X

  2. Mark these with a set of points, one for each point that meets X

  3. Example: (-4, 0), (-2, 0)

  4. Second value will always be zero because it is on the X or y=0 line. Mark these with parentheses because they are a set of points and not an interval

Y Intercept

  1. The Y intercept is anywhere the line crosses Y

  2. There will usually only be one place the line crosses Y, but there are instances of more. Mark with a point.

  3. Example: (0, 2)

Minimum (local)

  1. The Minimum (local) is anywhere the line dips into an upward opening parabola, or in other words, any time the line dips into a valley

  2. Mark Minimums with sets of points, marking the vertex (estimated usually) of the upward opening parabolas

  3. Example: (-1, 3), (4,2), (8, -1)

Maximum (local)

  1. The Maximum (local) is anywhere the line is at the peak of a downward facing parabola, or in other words, any time the line is at the top of a hill

  2. Mark Maximums with sets of points, marking the vertex of the downward opening parabolas

  3. Example: (-1, 3), (4, 2)

Minimum (absolute/global)

  1. The Global or Absolute Minimum is the lowest point on the line in the graph

Maximum (absolute/global)

  1. The Global or Absolute Maximum is the highest points on the line in the graph

Comparing Domain/Range to Maximum/Minimum

  1. After you have labeled all points, compare the maximum and minimum to the first and last range points.

  2. If the Maximum Y value is greater or equal to the second range point, that maximum is the absolute maximum

  3. If the Minimum Y value is less than or equal to the first range point, that minimum is the absolute minimum

Using Data to Answer Word Problems

  1. Use the Maximum to answer where the Maximum items will be sold, people will be happy, product will be most efficient etc.

  2. Use the Minimum to answer where the Minimum money could be lost, resource could be used, etc.

  3. Use the Domain to find the lowest and highest the X value COULD be

  4. Use the Range to find the lowest and highest the Y value COULD be

  5. Use the Constant to find where the production/money/ etc. is unchanging or stable