Unit 3 Graph Behavior

Increasing Function

A function in which for each x1 and x2, if x1<x2, then f(x1)<f(x2). As you look at the graph from left to right, the values of f(x) are increasing.

Decreasing Function

A function in which for each x1 and x2, if x1<x2, then f(x1)>f(x2). As you look at the graph from left to right, the values of f(x) are decreasing.

Odd Function

A function whose graph displays symmetry with respect to the origin is called an odd function. The points on the graph can be used to show that a function is an odd function. Both the x- and y-values have opposite signs. Thus, a given graph is the graph of an odd function if the value of the function when evaluated at -x is the opposite of the value when the function is evaluated for x. In other words, if, for all values of x, f(-x)=-f(x), then the function is an odd function.

Discontinuous Function

A function whose graph has a hole, jump, or vertical asymptote.

Continuous Function

A function whose graph is a single, unbroken curve.

Even Function

A function whose graph is symmetric with respect to the y-axis is called an even function. One way to show that a function is an even function is to analyze the points on the graph. The x-values have opposite signs because the points are on different sides of the y-axis, but the function values (y-values) are the same. Thus, a graph is the graph of an even function is the value of the function is the same for both x and -x for all values of x in the domain.

Horizontal Stretch and Compressions

A horizontal stretch of a function occurs when the function is stretched away from the y-axis. A horizontal compression of a function occurs when the function is compressed toward the y-axis. Multiplying the x-value in the function by a constant will result in either a horizontal stretch or compression, depending on the value of the constant. The graph of y=f(ax) is a horizontal stretch or compression of the graph of the function f. -If |a|>1, the graph is compressed towards the y-axis. -If 0<|a|<1, the graph is stretched away from the y-axis. Note that if a is negative the function is also being reflected across the y-axis.

Point of Discontinuity

A point where a function is discontinuous, or where the hole, jump, or vertical asymptote occurs.

X-Intercept

A point where the graph of a function intersects the x-axis.

Y-Intercept

A point where the graph of a function intersects the y-axis.

Reflections

A reflection across the x-axis occurs when the rule for a function is negated. Think about two points with opposite function values such as (1,2) and (1,-2). One is a reflection of the other across the x-axis. A reflection across the y-axis occurs when x is negated in the function rule. Think about two points with opposite x-values such as (-1,2) and (1,2). One is a reflection of the other across the y-axis.

Constant Function

A special case from the linear function family that is constant.

Vertical Stretches and Compressions

A vertical stretch of a function occurs when the function is stretched away from the x-axis. A vertical compression of a function occurs when the function is compressed towards the x-axis. Multiplying the original function by a constant will result in either a vertical stretch or compression, depending on the value of the constant. The graph of y=af(x) is a vertical stretch or compression of the graph f(x). -If |a|>1, the graph is streched away from the x-axis. -If 0<|a|<1, the graph is compressed toward the x-axis. Note that if a is negative, the function is reflected across the x-axis addition to the corresponding stretch for the positive value of a just mentioned.

Constant Function

A function in which for each x1 and x2, f(x1)=f(x2). As you look at the graph from left to right, the values of f(x) are the same.

Root Functions

Often referred to as radical functions, are functions of the form f(x)=^nsqrtx. Similar to power function, root functions differ based on whether the index is even or odd.

Absolute Value of a Function

The absolute value of a number is the distance the number is from 0. This can be represented by the following rule. |a|={a if a greater than or equal to 0, -a if a<0. For y=|f(x)| For any values of x for which f(x)greater than or equal to 0, the graph of y=|f(x)| is the same as the graph of f. For values of x for which f(x)<0, the graph of y=|f(x)| is a reflection of the graph of f across the x-axis. If you are given a graph of a function and asked to take its absolute value, the portion(s) of the graph on or above the x-axis will remain the same and the portion(s) of the graph below the x-axis will be reflected above the x-axis.

Vertical and Horizontal Asymptotes

The line x=a is a vertical asymptote of f(x) if f(x)(arrow)infinity or f(x)(arrow)-infinity as x(arrow)a. This means that if the function vallues are approaching infinity or negative infinity as x approaches a specific value, a, the line x=a is a vertical asymptote. The line y=b is a horizontal asymptote of the graph of f(x) if f(x)(arrow)b as x(arrow)infinity or x(arrow)-infinity. A horizontal asymptote occurs when the function values are approaching a specific value b, as x approaches infinity or negative infinity. Horizontal asymptotes are concerned with what is happening all the way to the left on a graph, as x approaches negative infinity, and all the way to the right on the graph, as x approaches infinity.

Subscript

The negative subscript indicates that x is approaching from the left. A positive indicates that x is approaching from the right. The vertical asymptote can also be indicated by the single statement f(x)(arrow)+/-infinity as x(arrow)(asymptote).

Transformations Order

1. Horizontal Shift 2. Stretch or compression 3. Reflection 4. Vertical Shift For stretches, compressions, and reflections, it does not matter whether you do them horizontally or vertically first.

Examples of Even and Odd Functions

1. f(x)=2x^4-3x^2+5 Begin by evaluating f(-x). f(-x)=2(-x)^4-3(-x)^2+5 =2x^4-3x^2+5 Observe that the result is the same as f(x). Thus, f(x) is an even function. 2. f(x)=-3x^3+4x-6 Begin by evaluating f(-x). f(-x)=-3(-x)^3+4(-x)-6 =3x^2-4x-6 Look closely at the resulting equation. The first two terms have signs that are opposite those of the corresponding terms of the original function, but the sign of the third term has stayed the same. Since the resulting function is not the same as the original and is not its opposite, this function is neither even nor odd. 3. f(x)=4x/x^2+3 Begin by evaluating f(-x). f(-x)0=4(-x)/(-x)^2+3 f(-x)0=-4x/x^2+3 f(-x)=-4x/x^2+3

Removable Discontinuity

A discontinuity at a point on the graph where the function is undefined or where the function value does not fit with the rest of the graph. Also called a point discontinuity. Graphically, it appears as a hole in the graph of the function.

Infinite Discontinuity (Type of Nonremovable Discontinuity)

A discontinuity that occurs at a vertical asymptote of a function. The function approaches infinity or negative infinity from each side of the asymptote. Also called an asymptotic discontinuity.

Jump Discontinuity (Type of Nonremovable Discontinuity)

A discontinuity where the value of the function jumps from one value to another that is not contiguous.

Transformation

A change made to the graph of a function by a horizontal or vertical shift, a reflection, or stretch or compression.

Horizontal Shift

A change made to the graph of a function by sliding the graph left or right. Adding a constant to the x-value in the original function will shift the graph either left or right, depending on the sign of the constant. The graph of y=f(x+a) is a horizontal shift of the graph of f(x). -If a is positive, the graph shifts left a units. -If a is negative, the graph shifts right a units.

Vertical Shift

A change made to the graph of a function by sliding the graph up or down. Adding a constant to the original function will shift the function either up or down, depending on the sign of the constant. The graph of y=f(x)+a is a vertical shift of the graph of f(x). -If a is positive, the graph shifts up a units. -If a is negative, the graph shifts down a units.

Asymptote

A line that the graph of a function approaches is called an asymptote. It is important to ntoe that an asymptote is not actually part of the graph of the function, but rather it provides information about the behavior of the function.

Infinity

An unbounded quantity that is greater than every real number and is represented by the symbol infinity. Negative infinity is less than every real number and is represented by -infinity. They symbol for the word approaches is an arrow.

Characteristics of the Constant Function: f(x)=c

Domain: (-infinity,infinity) Range: (c) Increasing Intervals: None Decreasing Intervals: None Constant Intervals: (-infinity,infinity) X-Intercept: None if c does not equal 0; if c=0, the function's graph is the x-axis Y-Intercept: (0,c) Even, Odd, Neither: Even; if c=0 the function is also odd Continuous or Discontinuous: Continuous Asymptotes: None End Behavior: f(x)(arrow)c as x(arrow)infinity; f(x)(arrow)c as x(arrow)-infinity.

Characteristics of the Parent Reciprocal Function: f(x)=1/x

Domain: (-infinity,0)U(0,infinity) Range: (-infinity,0)U(0,infinity) Increasing Intervals: None Decreasing Intervals: (-infinity,0)U(0,infinity) Constant Intervals: None X-Intercept: None Y-Intercept: None Even, Odd, Neither: Odd Continuous or Discontinuous: Discontinuous Asymptotes: x=0 and y=0 End Behavior: f(x)(arrow)0 as x(arrow)infinity; f(x)(arrow)0 as x(arrow)-infinity.

Characteristics of Odd-Index Root Functions: f(x)=^nsqrtx, where n is an odd integer greater than zero

Domain: (-infinity,infinity) Range: (-infinity,infinity) Increasing Intervals: (-infinity,infinity) Decreasing Interals: None Constant Intervals: None X-Intercept: (0,0) Y-Intercept: (0,0) Even, Odd, Neither: Odd Continuous or Discontinuous: Continuous Asymptotes: None End Behavior: f(x)(arrow)infinity as x(arrow)infinity; f(x)(arrow)-infinity as x(arrow)-infinity

Characteristics of Odd-Degree Power Functions: f(x)=x^n, where n is an odd integer greater than zero

Domain: (-infinity,infinity) Range: (-infinity,infinity) Increasing Intervals: (-infinity,infinity) Decreasing Intervals: None Constant Intervals: None X-Intercept: (0,0) Y-Intercept: (0,0) Even, Odd, Neither: Odd Continuous or Discontinuous: Continuous Asymptotes: None End Behavior: f(x)(arrow)infinity as x(arrow)infinity; f(x)(arrow)-infinity as x(arrow)-infinity

Characteristics of the Parent Linear Function: f(x)=x

Domain: (-infinity,infinity) Range: (-infinity,infinity) Increasing Intervals: (-infinity,infinity) Decreasing Intervals: None Constant Intervals: None X-Intercept: (0,0) Y-Intercept: (0,0) Even, Odd, Neither: Odd Continuous or Discontinuous: Continuous Asymptotes: None End Behavior: f(x)(arrow)infinity as x(arrow)infinity; f(x)(arrow)-infinnity as x(arrow)-infinity.

Characteristics of Exponential Functions: f(x)=b^x, b>1

Domain: (-infinity,infinity) Range: (0,infinity) Increasing Intervals: (-infinity,infinity) Decreasing Intervals: None Constant Intervals: None X-Intercept: None Y-Intercept: (0,1) Even, Odd, Neither: Neither Continuous or Discontinuous: Continuous Asymptotes: y=0 End Behavior: f(x)(arrow)infinity as x(arrow)infinity; f(x)(arrow)0 as x(arrow)-infinity

Characteristics of Exponential Functions: f(x)=b^x, 0<b<1

Domain: (-infinity,infinity) Range: (0,infinity) Increasing Intervals: None Decreasing Intervals: (-infinity,infinity) Constant Intervals: None X-Intercept: None Y-Intercept: (0,1) Even, Odd, Neither: Neither Continuous or Discontinuous: Continuous Asymptotes: y=0 End Behavior: f(x)(arrow)0 as x(arrow)infinity; f(x)(arrow)infinity as x(arrow)-inginity

Characteristics of the Parent Absolute Value Function: f(x)=|x|

Domain: (-infinity,infinity) Range: [0,infinity) Increasing Intervals: (0,infinity) Decreasing Intervals: (-infinity,0) Constant Intervals: None X-Intercept: (0,0) Y-Intercept: (0,0) Even, Odd, Neither: Even Continuous or Discontinuous: Continuous Asymptotes: None End Behavior: f(x)(arrow)infinity as x(arrow)infinity; f(x)(arrow)infinity as x(arrow)-infinity

Characteristics of Even-Degree Power Functions: f(x)=x^n, where n is an even integer greater than zero

Domain: (-infinity,infinity) Range: [0,infinity) Increasing Intervals: (0,infinity) Decreasing Intervals: (-infinity,0) Constant Intervals: None X-Intercept: (0,0) Y-Intercept: (0,0) Even, Odd, Neither: Even Continuous or Discontinuous: Continuous Asymptotes: None End Behavior: f(x)(arrow)infinity as x(arrow)infinity; f(x)arrow infinity as x(arrow)infinity

Characteristics of Even-Index Root Functions: f(x)=^nsqrtx, where n is an even integer greater than zero

Domain: [0,infinity) Range: [0,infinity) Increasing Intervals: (0,infinity) Decreasing Intervals: None Constant Intervals: None X-Intercept: (0,0) Y-Intercept: (0,0) Even, Odd, Neither: Neither Continuous or Discontinuous: Continuous Asymptotes: None End Behavior: f(x)(arrow)infinity as x(arrow)infinity

Characteristics of the Parent Logarithmic Function: f(x)=logb x, b>1

Domain:(0,infinity) Range: (-infinity,infinity) Increasing Intervals: (0,infinity) Decreasing Intervals: None Constant Intervals: None X-Intercept: (1,0) Y-Intercept: None Even, Odd, Neither: Neither Continuous or Discontinuous: Continuous Asymptotes: x=0 End Behavior: f(x)(arrow)infinity as x(arrow)infinity; f(x)(arrow)-infinity as x(arrow)0

Characteristics of the Parent Logarithmic Function: f(x)=logb x, 0<b<1

Domain:(0,infinity) Range: (-infinity,infinity) Increasing Intervals: None Decreasing Intervals: (0,infinity) Constant Intervals: None X-Intercept: (1,0) Y-Intercept: None Even, Odd, Neither: Neither Continuous or Discontinuous: Continuous Asymptotes: x=0 End Behavior: f(x)(arrow)-infinity as x(arrow)infinity; f(x)(arrow)infinity as x(arrow)0

Parent Reciprocal Function

Function f(x)=1/x. For this function, if x is positive and increases, the value of the function decreases, producing fractions that get closer and closer to zero. So, as the positive x-value are approaching infinity, the function values are approaching zero. As the positive x-value decrease, approaching zero from the right, the value of the fraction increases. These patterns are related when x is negative. For negative x-values, the function approaches zero as x approaches negative infinity, and the function approaches negative infinity as x approaches zero. If x=0, since you cannot divide by zero, the function is undefined at x=0.

Exponential Function

Functions of the form f(x)=b^x. Graphs of exponential functions differ depending on the value of the base. The base of an exponential function can be any positive real number except for 1. When the base is greater than 1, the function behave differently than when the base is between 0 and 1.

Logarithmic Functions

Functions of the form f(x)=logb x. The function logb x is the power to which b is raised to get x. Similar to exponential functions, logarithmic functions depend on the identified base. It is important to note that the base of a logarithm must be a positive real number other than 1. The two cases of logarithmic functions occur when the base is greater than 1 and when it is between 0 and 1.

Family of Functions

Functions whose equations have a similar form and whose graphs have the same basic shape.

End Behavior

How the function behaves at both ends of its domain. Horizontal asymptotes are specific types of end behavior; that is, when the function approaches a aspecific value as x approaches infinity or negative infinity. However, the function could also increase or decrease without bound at either end. Cubic Function - The end behavior can be described symbolically: f(x)(arrow)infinity as x(arrow)infinity, and f(x)(arrow)-infinity as x(arrow)-infinity.

Parent Linear Function

Identity function.

Even-Index Root Functions

If the index is even, then the radican, or value inside the radical, must begreater than or equal to zero. This restricts the domain to non-negative values. The graph will start at the origin, since any root of zero will be zero. As the x-values increase, so do the function values.

Logarithmic Function Greater Than 1

If your function values are the powers of the base necessary to get the x-value, it follows that as the x-values increase, the powers of the base, or function values, also increase. If the base is greater than 1, then any power of the base will result in positive x-values.

Base of Function Between 0 and 1

Like the last example, raising a fraction to a negative exponent will require taking the reciprocal of the fraction and raising it to a positive exponent. As the exponents get more and more negative, the function values are going to increase. The opposite is going to be true for raising a fractoin to increasing positive exponents. The function values will decrease. The function is now decreasing instead of increasing. This also changes the end behavior.

Power Functions

Power functions are functions of the form f(x)=x^n. The graph of a power function looks different depending on whether the degree is even or odd.

Negative Exponent

Raising a value to a negative exponent involves moving the value to the denominator and raising it to a positive exponent. For example, f(-2)=2^-2=2/2^2=2/4. As x-value are decreasing, the function values are becoming smaller fractions. As the x-values increase the function values also increase. The function values are going to increase much more quickly than the x-values when x is greater than zero.

Even Power Functions

Raising any non-zero value to an even number results in a positive number. Furthermore, the function values will be equal for input values that are opposites in each other. For example, when graphing the quadratic power function f(x)=x^2, f(3)=f(-3)=9 since (-3)^2=(3)^2.

Odd Power Functions

Since raising negative numbers to odd exponents results in negative number, the range of these functions will include negative numbers. Input values that are opposites of each other will produce opposite function values. For example, consider f(x)=x^3. Observe that, for any value of a, the function at -a is the opposite of the function value at a: f(-a)=(-a)^3=-a^3=-f(a). The function values will be decreasing as the x-values are decreasing and will be increasing as the x-values are increasing. The one point that the graph of all power functions will contain, regardless of whether the exponent is even or odd, is (0,0) since zero raised to any exponent (other than zero) is zero.

Subscripts

Subscripts can be used to differentiate functions.

Odd-Index Root Functions

The biggest difference is that the domain is no longer restricted to non-negative numbers. If the index is odd, the radicand can be any real number. The graph will still pass through the origin, as any root of zero is zero. Input values that are opposites of each other will produce opposite function values. For example, look at the function f(x)=^3sqrtx: f(8)=^3sqrt8=2 and f(-8)=^3sqrt-8=^3sqrt-8=-2.

Parent Absolute Value Function

The function f(x)=|x|. Remember that the absolute value of any number is its distance from zero. The function values will always be nonnegative. The absolute value parent function looks like the letter v centered at the origin.

Checking Results

The result can also be verified by creating a table of values and checking it against the graph of g(x).

Parent Function

The simplest function of a family of functions is called the parent function. All other functions in the family represent some transformation of the parent function.

Logarithmic Function Between 0 and 1

They are similar to the graphs of logarithmic function when b>1 except they are reflected over the x-axis.

Test for Even or Odd Functions

To test if a function f(x) is even, odd, or neither, substitute -x for x and simplify. -If f(-x)=f(x), then the function is even. -If f(-x)=-f(x), then the function is odd. -Otherwise, the function is neither even nor odd.

Linear Function Family Members

Variations on the parent linear function can be represented by the general equation f(x)=ax+b, where a and b are any real numbers. For the parent linear function (the identity function), a=1 and b=0. For the constant function, a=0 and b=c. Consider the functions in this family other than the identity and constant functions. Depending on the values of a and b, these functions can have some different characteristics than the parent, even though their basic shape is the same. -Negative values of a produce functions that are decreasing rather tahn increasing. -Any nonzero value of b will change the x- and y- intercepts. The special case of the constant function when c=0 is mentioned in its key concept. -There will be an infinite number of x-intercepts since the resulting graph will actually be the x-axis. -The function will be odd as well as even.