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Math  /  Algebra

QuestionFind a possible formula for the function graphed below. The xx-intercepts are marked with points located at (5,0)(5,0) and (−4,0)(-4,0), while the yy-intercept is marked with a point located at (0,−53)\left(0,-\frac{5}{3}\right). The asymptotes are y=−1,x=−3y=-1, x=-3, and x=4x=4. Give your formula as a reduced rational function. f(x)=f(x)= □\square help (formulas) (Click on graph to enlarge)

Studdy Solution

STEP 1

1. The function is a rational function with vertical asymptotes at x=−3 x = -3 and x=4 x = 4 .
2. The horizontal asymptote is at y=−1 y = -1 .
3. The x x -intercepts are at (5,0) (5, 0) and (−4,0) (-4, 0) .
4. The y y -intercept is at (0,−53) \left(0, -\frac{5}{3}\right) .

STEP 2

1. Determine the form of the rational function based on asymptotes.
2. Incorporate x x -intercepts into the function.
3. Use the y y -intercept to find the constant factor.
4. Write the final reduced rational function.

STEP 3

The vertical asymptotes at x=−3 x = -3 and x=4 x = 4 suggest that the denominator of the rational function has factors (x+3) (x + 3) and (x−4) (x - 4) .
The horizontal asymptote at y=−1 y = -1 indicates that the degrees of the numerator and denominator are the same, and the leading coefficient of the numerator divided by the leading coefficient of the denominator is −1-1.

STEP 4

The x x -intercepts at (5,0) (5, 0) and (−4,0) (-4, 0) suggest that the numerator has factors (x−5) (x - 5) and (x+4) (x + 4) .

STEP 5

The rational function can be expressed as:
f(x)=a(x−5)(x+4)(x+3)(x−4) f(x) = \frac{a(x - 5)(x + 4)}{(x + 3)(x - 4)}
To find a a , use the y y -intercept (0,−53)\left(0, -\frac{5}{3}\right):
f(0)=a(0−5)(0+4)(0+3)(0−4)=−53 f(0) = \frac{a(0 - 5)(0 + 4)}{(0 + 3)(0 - 4)} = -\frac{5}{3}
a(−5)(4)3(−4)=−53 \frac{a(-5)(4)}{3(-4)} = -\frac{5}{3}
−20a−12=−53 \frac{-20a}{-12} = -\frac{5}{3}
5a3=−53 \frac{5a}{3} = -\frac{5}{3}
5a=−5 5a = -5
a=−1 a = -1

STEP 6

Substitute a=−1 a = -1 back into the function:
f(x)=−(x−5)(x+4)(x+3)(x−4) f(x) = \frac{-(x - 5)(x + 4)}{(x + 3)(x - 4)}
Simplify the numerator:
f(x)=−(x2−x−20)(x+3)(x−4) f(x) = \frac{-(x^2 - x - 20)}{(x + 3)(x - 4)}
f(x)=−x2+x+20(x+3)(x−4) f(x) = \frac{-x^2 + x + 20}{(x + 3)(x - 4)}
This is the reduced rational function.
The possible formula for the function is:
f(x)=−x2+x+20(x+3)(x−4) f(x) = \frac{-x^2 + x + 20}{(x + 3)(x - 4)}

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