Math  /  Algebra

QuestionWrite an equation for a rational function with:
Vertical asymptotes at x=2x=-2 and x=2x=2 xx intercepts at x=3x=3 and x=6x=6 yy intercept at 5 y=y=

Studdy Solution

STEP 1

1. The rational function has vertical asymptotes at x=2 x = -2 and x=2 x = 2 .
2. The rational function has x x -intercepts at x=3 x = 3 and x=6 x = 6 .
3. The rational function has a y y -intercept at y=5 y = 5 .

STEP 2

1. Determine the form of the rational function.
2. Incorporate the vertical asymptotes into the function.
3. Incorporate the x x -intercepts into the function.
4. Determine the constant to satisfy the y y -intercept condition.
5. Write the final equation of the rational function.

STEP 3

Determine the form of the rational function.
A rational function can be expressed as:
y=N(x)D(x) y = \frac{N(x)}{D(x)}
where N(x) N(x) and D(x) D(x) are polynomials.

STEP 4

Incorporate the vertical asymptotes into the function.
Vertical asymptotes occur where the denominator is zero. Therefore, the denominator should include factors corresponding to x=2 x = -2 and x=2 x = 2 :
D(x)=(x+2)(x2) D(x) = (x + 2)(x - 2)

STEP 5

Incorporate the x x -intercepts into the function.
x x -intercepts occur where the numerator is zero. Therefore, the numerator should include factors corresponding to x=3 x = 3 and x=6 x = 6 :
N(x)=(x3)(x6) N(x) = (x - 3)(x - 6)

STEP 6

Determine the constant to satisfy the y y -intercept condition.
The y y -intercept occurs when x=0 x = 0 . Substitute x=0 x = 0 into the function and set it equal to 5:
y=(03)(06)(0+2)(02)=184=92 y = \frac{(0 - 3)(0 - 6)}{(0 + 2)(0 - 2)} = \frac{18}{-4} = -\frac{9}{2}
To adjust the function to have a y y -intercept of 5, introduce a constant k k :
5=k×92 5 = k \times -\frac{9}{2}
Solve for k k :
k=592=109 k = \frac{5}{-\frac{9}{2}} = -\frac{10}{9}

STEP 7

Write the final equation of the rational function.
Substitute k k back into the function:
y=109×(x3)(x6)(x+2)(x2) y = -\frac{10}{9} \times \frac{(x - 3)(x - 6)}{(x + 2)(x - 2)}
This is the equation of the rational function with the given conditions.

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