The features of a function graph can show us many aspects of the relationship represented by the function. Let's take a look at the more popular graphical features.

intercepts
Intercepts are the locations (points) where the graph crosses (or touches) either the x-axis or y-axis.
• To find the y-intercept, look where x = 0.
Remember: the y-intercept will have an x-coordinate of 0.
y = -2x + 2
y = -2(0) + 2;    y = 2     y-intercept: (0,2)
(You can also read the y-intercept, b, from the function equation
if it is in y = mx + b form.)


• To find the x-intercept, look where y = 0.
Remember: the x-intercept will have a y-coordinate of 0.
y = -2x + 2
0 = -2x + 2;     2x = 2;     x = 1    
x-intercept: (1,0)
fgraoh1

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posneg notzero

y-values positive
or
y-values negative

porneggraph1 • The positive regions of a function are those intervals where the function is above the x-axis.
It is where the y-values are positive (not zero).

• The negative regions of a function are those intervals where the function is below the x-axis.
It is where the y-values are negative (not zero).

y-values that are on the x-axis are neither positive nor negative. The x-axis is where y = 0.

Some functions are positive over their entire domain
(All y-values above the x-axis.)
posgraph1
positive: -∞ < x < +∞
or "all Reals", or (-∞,+∞)
Some functions are negative over their entire domain.
(All y-values below the x-axis.)
posgraph2
negative: -∞ < x < +∞
or "all Reals", or (-∞,+∞)
Some functions have both positive and negative regions.
(y-values above and below x-axis)
posgraph3
positive: x > 0 or (0,+∞)
negative: x < 0 or (-∞,0)
(do not include zero)

hintgal
Secret to Finding the Intervals!
The secret to correctly stating the intervals where a function is positive or negative is to remember that the intervals ALWAYS pertain to the locations of the x-values. Think of reading the graph from left to right along the x-axis.
Do NOT read numbers off the y-axis for the intervals. Stay on the x-axis!

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incdec  

When looking for sections of a graph that are increasing or decreasing, be sure to look at (or "read") the graph from left to right.

Increasing: A function is increasing, if as x increases (reading from left to right), y also increases .

In plain English, as you look at the graph, from left to right, the graph goes up-hill. The graph has a positive slope.

Example: The function (graph) at the right is increasing from the point (-5,-3) to the point (-2,1), which is described as increasing when -5 < x < -2 .

It also increases from the point (1,1) to the point (3,4), described as increasing when 1 < x < 3.

fgraph2

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Decreasing: A function is decreasing, if as x increases (reading from left to right), y decreases. In plain English, as you look at the graph, from left to right, the graph goes down-hill. The graph has a negative slope.

Example: The graph shown above is decreasing from the point (3,4) to the point (5,-5), described as decreasing when 3 < x < 5.

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Constant: A function is constant, if as x increases (reading from left to right), y stays the same. In plain English, as you look at the graph, from left to right, the graph goes flat (horizontal). The graph has a slope of zero.

Example: The graph shown above is constant from the point (-2,1) to the point (1,1), described as constant when -2 < x < 1. The y-values of all points in this interval are "one".

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hintgal
Intervals of increasing, decreasing or constant ALWAYS pertain to x-values.
Do NOT read numbers off the y-axis.
Stay on the x-axis for these intervals!


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