Characteristics of Functions

x and y intercepts The x intercept is where the graph crosses the x axis. The y intercept is where the graph crosses the y axis. x intercepts

x and y intercepts The x intercept is a point in the equation where the y-value is zero. Example: Find the x intercepts of Set y =0, solve for x. x intercepts:

x and y intercepts The y intercept is a point in the equation where the x-value is zero. Example: Find the y intercepts of Set x =0, solve for y. y intercepts: (0,-2) (0,2)

Increasing, Decreasing, Constant Intervals A function f is increasing on an interval if as x increases, then f(x) increases. A function f is decreasing on an interval if as x increases, then f(x) decreases. A function f is constant on an interval if as x increases, then f(x) remains the same.

Increasing, Decreasing, Constant Intervals A function f is increasing on an interval if as x increases, then f(x) increases. A function f is decreasing on an interval if as x increases, then f(x) decreases. f(x) is decreasing in the interval . vertex (1.5,-2) f(x) is increasing in the interval .

Increasing, Decreasing, Constant Intervals A function f is increasing on an interval if as x increases, then f(x) increases. A function f is decreasing on an interval if as x increases, then f(x) decreases. A function f is constant on an interval if as x increases, then f(x) remains the same. Find the interval(s) over which the interval is increasing, decreasing and constant? Answer Now

Increasing, Decreasing, Constant Intervals A function f is increasing on an interval if as x increases, then f(x) increases. A function f is decreasing on an interval if as x increases, then f(x) decreases. A function f is constant on an interval if as x increases, then f(x) remains the same. f(x) is decreasing in the interval (-1,1). f(x) is increasing in the intervals

Increasing, Decreasing, Constant Intervals A function f is increasing on an interval if as x increases, then f(x) increases. A function f is decreasing on an interval if as x increases, then f(x) decreases. A function f is constant on an interval if as x increases, then f(x) remains the same. Find the interval(s) over which the interval is increasing, decreasing and constant? Answer Now

Increasing, Decreasing, Constant Intervals A function f is increasing on an interval if as x increases, then f(x) increases. A function f is decreasing on an interval if as x increases, then f(x) decreases. A function f is constant on an interval if as x increases, then f(x) remains the same. f(x) is decreasing over the intervals f(x) is increasing over the interval (3,5). f(x) is constant over the interval (-1,3).

Minimum/Maximum A min/max is where there is a "turning point" on a graph . It is a point at which the graph changes from increasing to decreasing (it looks like a "hill"), or from decreasing to increasing (it looks like a "valley").

These turning points are referred to as relative (or local) maxima or minima. The designation of "relative" (or "local") tells you that this point may not be the largest (or smallest) value reached by this function. It is only a maximum (or minimum) "relative" to a small interval of the function surrounding this point. The one, true, largest (or smallest) value reached by the entire function is called the absolute maximum (or minimum), or the global maximum (or minimum).

End Behavior of Functions The end behavior of a graph describes the far left and the far right portions of the graph. Using the leading coefficient and the degree of the polynomial, we can determine the end behaviors of the graph. This is often called the Leading Coefficient Test.

End Behavior of Functions First determine whether the degree of the polynomial is even or odd. degree = 2 so it is even Next determine whether the leading coefficient is positive or negative. Leading coefficient = 2 so it is positive

Leading Coefficient: + END BEHAVIOR Degree: Even Leading Coefficient: + End Behavior: Up Up So we say f(x) ∞ , as x ∞ and f(x) ∞ , as x - ∞

End Behavior: Down Down Degree: Even Leading Coefficient: End Behavior: Down Down So we say f(x) - ∞ , as x ∞ and f(x) - ∞ , as x - ∞

Leading Coefficient: + END BEHAVIOR Degree: Odd Leading Coefficient: + End Behavior: Down Up So we say f(x) ∞ , as x ∞ and f(x) - ∞ , as x - ∞

Degree: Odd Leading Coefficient: End Behavior: Up Down END BEHAVIOR So we say f(x) - ∞ , as x ∞ and f(x) ∞ , as x - ∞

END BEHAVIOR PRACTICE Give the End Behavior:

END BEHAVIOR PRACTICE Give the End Behavior: Up Down Up Up Down Up Down Down

Symmetry Some graphs have an axis of symmetry over which the graph becomes a reflection (or mirror image) of itself.

Horizontal Symmetry If a graph possesses horizontal symmetry (such as a reflection over the x-axis), the graph is not a function. possesses both (x,y) and (x,-y)

Vertical Symmetry A function graph may possess vertical symmetry (such as a reflection over the y-axis). possesses both (x,y) and (-x,y)

About the origin Symmetry A function graph may possess symmetry about the origin (0,0). This is the same as a rotation of 180º  possesses both (x,y) and (-x,-y).

Rate of Change Recall formula for slope of a line through two points For any function we could determine the slope for two points on the graph This is the average rate of change for the function on the interval from x1 to x2

Rate of Change Jasper has invested an amount of money into a savings account. The graph to the right shows the value of his investment over a period of time. What is the rate of change for the interval [1, 3]?

Rate of Change Find the average rate of change of f(x) = 2x2 – 3 when x1= 2 and x2 = 4