To write a rational function, define a variable for the number of additional free throws needed.
6 free throws
Practice makes perfect
We know that a basketball player made 21 of her last 30 free throws. In other words, she made 70 % of the last free throws.
21/30= 0.70
We want to find the number of additional free throws needed in order to increase this percentage to 75 %. Let x be the number of additional free throws needed. We need to add x to both the numerator and the denominator.
21+ x/30+ x
Let it be equal to y, and it will model the player's free throw percentage as a rational function.
y=21+ x/30+ x
Let's take a look at its graph and the graph of the function y = 0.75.
The rational function and the function y=0.75 intersect at the point (6,0.75). Therefore, she needs to score 6 consecutive free throws in order to increase this percentage to 75 %.
Showing Our Work
Drawing the Graph of the Rational Function
To graph the rational function, we will follow three steps.
Since the degrees of the numerator and the denominator are the same, the horizontal asymptote is the ratio of the coefficient of the term of greatest degree in the numerator and the denominator.
y=21+( 1)x/30+( 1)x ⇒ y=1/1= 1
Therefore, the horizontal asymptote is y=1. We see that - 30 is the zero of the denominator but not a zero of the numerator. This means that x=- 30 is the vertical asymptote. y=21+x/30+x l → x= - 30
Let's draw the asymptotes.
Plotting Some Points
Let's find some points both to the left and the right of the vertical asymptote.
x
21+x/30 +x
y=21+x/30 +x
Left of the Asymptote
- 60
21 + ( - 60)/30+ ( - 60)
1.3
- 50
21 + ( - 50)/30+ ( - 50)
1.45
- 40
21 + ( - 40)/30+ ( - 40)
1.9
Right of the Asymptote
- 20
21 + ( - 20)/30+ ( - 20)
0.1
- 10
21 + ( - 10)/30+ ( - 10)
0.55
0
21 + ( 0)/30+ ( 0)
0.7
Let's plot the (x, y) points so we can see the behavior of the function.
Sketching The Graph
Finally, we will use the points to sketch the graph. It must approach both the horizontal and vertical asymptotes.
