Understanding Key Characteristics of Rational Function Graphs

​Understanding Rational Function Key Characteristics and Graphs

• I can determine the vertical and horizontal asymptotes when given a function.

• I can determine if a functions has a hole.

• I can use the key characteristics to sketch a graph of the function.

Multiple Choice

Select ALL the information that applies to the given rational function.
$$f\left(x\right)=\frac{x-3}{-2x^2+8x-6}$$  

• Vertical Asymptotes. at x = 1, x = 3

• No Holes

• Horizontal Asymptote. at y = 0

• y-intercept at (0, -1/2)

Explanation Slide...

The horizontal asymptote at y = 0 is correct because the degree of the numerator is less than the degree of the denominator. The other options are incorrect as the vertical asymptotes and y-intercept do not match the function.

Multiple Select

Select ALL the information that applies to the given rational function. $$f\left(x\right)=\frac{2x^2-18}{x^2-4x+3}$$  

• Vertical Asymptote. at x = -3 and x = 1

• Hole at (3, 6)

• Horizontal Asymptote at y = 2

• y-intercept at (-6,0)

Explanation Slide...

The hole at (3, 6) occurs because the numerator and denominator share a factor that cancels out. The horizontal asymptote at y = 2 is determined by the leading coefficients of the numerator and denominator.

Multiple Select

Select ALL the information that applies to the given rational function. $$f\left(x\right)=\frac{x^2-x}{-4x^2+16}$$  

• Vertical Asymptote at x = -2

• No Holes

• Horizontal Asymptote. at y = -1/4

• x-intercepts at (0,0) and (1,0)

Explanation Slide...

The function has no holes since the numerator and denominator have no common factors. The vertical asymptote is at x = -2 (denominator = 0). The horizontal asymptote is y = -1/4 (leading coefficients). The x-intercepts are (0,0) and (1,0).

Multiple Choice

Select ALL the information that applies to the given rational function. $$f\left(x\right)=\frac{x+3}{-3x+6}$$  

• Vertical Asymptote at x = -2

• No Holes

• Horizontal Asymptote at y = 1/3

• y-intercepts at (0, -1/2)

Explanation Slide...

The function has no holes since the numerator and denominator do not share any common factors. The vertical asymptote is at x = 2, not -2, and the horizontal asymptote is y = -1/3, not 1/3. The y-intercept is (0, 1).

Multiple Select

Which statements are true for the given rational function. Select all that apply.

• The horizontal asymptote is at y = 0.

• There is a hole at (6, ¾ ).

• The vertical asymptote is at x = -2.

• The x and y intercepts are at the origin.

• The domain is all real numbers except -2.

Explanation Slide...

The hole at (6, ¾) indicates a removable discontinuity. The vertical asymptote at x = -2 shows where the function is undefined. The x and y intercepts at the origin confirm the function crosses both axes there.

Multiple Choice

 Graph the function.

$$f\left(x\right)=\frac{-2x^2+2x+24}{x^3-x^2-12x}$$

Multiple Choice

Match the graph to the correct rational equation below.

• $$f(x)=\frac{x^2}{x^2+x-12}$$

• $$f(x)=\frac{4x^2}{x^2-x-12}$$

• $$f(x)=\frac{2x^2-2}{2x^2}$$

• $$f(x)=\frac{x}{x^2+x-12}$$

Multiple Choice

Graph the function.

Multiple Choice

Which function is represented by this graph?

• $$\frac{x}{x^2-9}$$  
  

• $$\frac{x^2+9}{x}$$  
  

• $$\frac{x+3}{x^2-9}$$  

• $$\frac{x^2+3}{x^2-9}$$  

Multiple Choice

What are the equations of the asymptotes?

• x=1, x= 2, y =1, y= 2

• x= 2, x=-2, y = 1

• x=2 y =-1

• x=1 y =2, y =-2

Explanation Slide...

The correct asymptotes for the given function are x=2 and x=-2, which indicate vertical asymptotes, and y=1, indicating a horizontal asymptote. Thus, the correct choice is x=2, x=-2, y=1.

Multiple Choice

Which asymptote(s) are determined by setting the denominator equal to zero?

• vertical

• horizontal

• end behavior

• none

Explanation Slide...

Vertical asymptotes occur when the denominator of a rational function is set to zero, leading to undefined values. Therefore, the correct answer is vertical.

Multiple Choice

Find the coordinates of the hole.

• (-3, -3)

• (-3, 3)

• (3, -3)

• (3, 3)

Explanation Slide...

The hole in a function occurs when there is a common factor in both the numerator and denominator.
factor both the numerator and denominator, identify any common factors, set the common factor equal to zero to find the x-coordinate of the hole, and then substitute that x-value into the simplified function to find the y-coordinate. 

Multiple Choice

What is the domain?

• All real numbers

• All real numbers except 4

• All real numbers except 2

• All real numbers except 2 and -2

Explanation Slide...

The domain excludes values that make the function undefined. Set the denominator equal to zero to find the excluded values.

Multiple Choice

What is the end behavior as x approaches negative infinity?

• y approaches negative infinity

• y approaches positive infinity

• y approaches 1

• y approaches 2

Explanation Slide...

As x approaches negative infinity, the function stabilizes and approaches a constant value. In this case, y approaches 1, indicating that the end behavior is horizontal and does not diverge to infinity.

Multiple Choice

True or False?

Rational functions can have more than one vertical asymptote.

• True, each vertical asymptote is obtained by setting the non-canceled factors of the denominator = 0.

• True, each vertical asymptote is obtained by setting the factors of the numerator = 0.

• False, there can be only one determined by the power of the numerator compared to the power of the denominator.

Explanation Slide...

True. Rational functions can have multiple vertical asymptotes, which occur at the values of x that make the non-canceled factors of the denominator equal to zero.

Multiple Choice

True or False?

The graph of a rational function can cross a horizontal asymptote.

• True, a graph can cross a horizontal asymptote because it really dictates the end behavior.

• False.

Explanation Slide...

True. A graph of a rational function can cross a horizontal asymptote, as the asymptote describes the end behavior but does not restrict the function's values in the middle.

Multiple Choice

Holes and Vertical Asymptotes are points of discontinuity.

• True

• False

Explanation Slide...

True. Holes and vertical asymptotes indicate points where a function is not defined, making them types of discontinuities in the graph of the function.

Multiple Choice

To find the y-intercepts, plug in zero for x, let x = 0; then evaluate f(0).

• True

• False

Multiple Choice

You can have more than one vertical asymptote.

• True

• False

Explanation Slide...

True. A function can have multiple vertical asymptotes, typically occurring at values where the function approaches infinity due to division by zero in its rational form.

Multiple Choice

To find the x-intercept, you make the denominator = 0.

• True

• False

Explanation Slide...

The statement is false. To find the x-intercept, you set the numerator equal to zero, not the denominator. The denominator being zero indicates vertical asymptotes, not x-intercepts.

Multiple Choice

There is no hole if you can't factor or cancel a factor.

• True

• False

Explanation Slide...

The statement is true because a hole in a function occurs when a factor can be canceled. If you cannot factor or cancel, there is no hole present in the function.

Multiple Choice

The horizontal asymptote is y= 0 when:

• the degree of the numerator and denominator are equal

• the degree of the numerator is less than the degree of the denominator

• the degree of the numerator is greater than the degree of the denominator

• the numerator equals zero

Explanation Slide...

The horizontal asymptote is y=0 when the degree of the numerator is less than the degree of the denominator. This means that as x approaches infinity, the value of the function approaches zero.

Multiple Choice

There is a slant (oblique) asymptote when:

• the degree of the numerator and denominator are equal

• the degree of the numerator is less than the degree of the denominator

• the degree of the numerator is greater than the degree of the denominator

• the numerator equals zero

Explanation Slide...

A slant (oblique) asymptote occurs when the degree of the numerator is greater than the degree of the denominator. This results in a linear function that describes the behavior of the rational function as it approaches infinity.

Multiple Choice

The horizontal asymptote is equal to the coefficients of x when:

• the degree of the numerator and denominator are equal

• the degree of the numerator is less than the degree of the denominator

• the degree of the numerator is greater than the degree of the denominator

• the numerator equals zero

Explanation Slide...

The horizontal asymptote is determined by the leading coefficients when the degrees of the numerator and denominator are equal. In this case, the horizontal asymptote is the ratio of these coefficients.

Multiple Choice

Which intercept(s) are determined by setting the numerator equal to zero ?

• vertical

• horizontal

• x

• y

Explanation Slide...

The x-intercepts are found by setting the numerator of a rational function to zero. This gives the points where the graph crosses the x-axis, making 'x' the correct choice. Vertical and horizontal intercepts are determined differently.

Multiple Choice

How do we determine the vertical asymptotes of a rational function?

• Find the Excluded values

• Degrees of the numerator and denominator

• Set the numerator equal to 0

• Set the simplified numerator equal to 0

Explanation Slide...

To determine vertical asymptotes of a rational function, we find the excluded values by setting the denominator equal to zero. These values indicate where the function is undefined, leading to vertical asymptotes.

Multiple Choice

To find the y-intercepts, plug in zero for y, let y = 0; then evaluate.

• True

• False

Explanation Slide...

To find the y-intercepts, you set x = 0, not y. Therefore, the statement is false.

Multiple Choice

True or False?

Rational functions can have more than one horizontal asymptote.

• True, each vertical asymptote is obtained by setting the non-canceled factors of the denominator = 0.

• True, each vertical asymptote is obtained by setting the factors of the numerator = 0.

• False, there can be only one determined by the power of the numerator compared to the power of the denominator.

Explanation Slide...

False, a rational function can only have one horizontal asymptote, which is determined by comparing the degrees of the numerator and denominator. This asymptote reflects the end behavior of the function.

Multiple Select

x = 0 is the equation of the ____ axis

• x

• y

• horizntal

• vertical

Explanation Slide...

The equation x = 0 represents the vertical axis in a Cartesian coordinate system, where all points have an x-coordinate of 0. Therefore, the correct answers are 'y' for the y-axis and 'vertical' for the orientation.

Multiple Select

y = 0 is the equation of the ____ axis

• x

• y

• horizntal

• vertical

Explanation Slide...

The equation y = 0 represents the x-axis in a Cartesian coordinate system, which is a horizontal line where the y-coordinate is always zero. Thus, the correct answers are 'x' and 'horizontal'.

Multiple Select

$$a>0$$  means

• $$a\ is\ positive$$  

• $$a\ is\ negative$$  

• curve is decreasing

• curve is increasing

Explanation Slide...

The statement a > 0 indicates that 'a' is positive. When 'a' is positive, it implies that the curve is increasing, as positive values typically correspond to an upward trend in mathematical functions.

Multiple Select

$$a<0$$  means

• $$a\ is\ positive$$  

• $$a\ is\ negative$$  

• curve is increasing

• curve is decreasing

Explanation Slide...

The statement a<0 indicates that a is negative. When a is negative, the curve associated with it is decreasing, as negative values typically lead to a downward trend.

Open Ended

Explain the difference between a rational expression and a rational function.

Explanation Slide...

A rational expression is a fraction where both the numerator and denominator are polynomials. A rational function is a specific type of function defined by a rational expression, which can be evaluated for different values of the variable.

Open Ended

Define discontinuity

Explanation Slide...

Discontinuity refers to a break or gap in a function or sequence, where it is not continuous. This can occur in mathematics when a function does not have a defined value at a certain point, leading to an interruption in its graph.

Multiple Choice

How do we determine the vertical asymptotes of a rational function?

• Simplify by factoring

• Degrees of the numerator and denominator

• Set the denominator equal to 0

• Set the simplified numerator equal to 0

Explanation Slide...

To find vertical asymptotes of a rational function, we set the denominator equal to 0. This identifies the values where the function is undefined, which correspond to the vertical asymptotes.