Question(2X+)2(X->) f(x) = (2x+1)² (X<>) (X+7)085) FIN) (X+1) FIN 15 HA -ZEE VA ROOTS (x+7) -MUL HOLES SIGN CHART -7-√2 -2-19 SKETCH X+7++ -ENT - - SKE

Studdy Solution

STEP 1

1. The function given is $f(x) = \frac{(2x+1)^2(x-7)}{(x+7)(x-2)^2}$ .
2. We need to find horizontal asymptotes (HA), vertical asymptotes (VA), roots (zeros), and holes of the function.
3. We will use the sign chart to determine the behavior of the function across different intervals.
4. The function is a rational function, which means it is a ratio of polynomials.

STEP 2

1. Identify and simplify the function.
2. Find the horizontal asymptotes.
3. Find the vertical asymptotes.
4. Determine the roots of the function.
5. Identify any holes in the function.
6. Construct a sign chart.
7. Sketch the function.

STEP 3

The function is already given in a simplified form:
$f(x) = \frac{(2x+1)^2(x-7)}{(x+7)(x-2)^2}$

STEP 4

To find the horizontal asymptotes, compare the degrees of the numerator and the denominator:
- The degree of the numerator is $3$ (from $(2x+1)^2(x-7)$ ). - The degree of the denominator is $3$ (from $(x+7)(x-2)^2$ ).
Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
$\text{Horizontal Asymptote: } y = \frac{4}{1} = 4$

STEP 5

Vertical asymptotes occur where the denominator is zero but the numerator is not zero:
- Set the denominator equal to zero: $(x+7)(x-2)^2 = 0$ . - Solve for $x$ : $x = -7$ and $x = 2$ .
Check the numerator at these points: - At $x = -7$ , the numerator $(2(-7)+1)^2(-7-7)$ is not zero. - At $x = 2$ , the numerator $(2(2)+1)^2(2-7)$ is not zero.
Thus, vertical asymptotes are at $x = -7$ and $x = 2$ .

STEP 6

Roots (zeros) occur where the numerator is zero and the denominator is not zero:
- Set the numerator equal to zero: $(2x+1)^2(x-7) = 0$ . - Solve for $x$ : $2x+1 = 0$ or $x-7 = 0$ .
For $2x+1 = 0$ , $x = -\frac{1}{2}$ . For $x-7 = 0$ , $x = 7$ .
Thus, the roots are $x = -\frac{1}{2}$ and $x = 7$ .

STEP 7

Holes occur where both the numerator and denominator are zero:
- Check the common factors in numerator and denominator. There are no common factors, so there are no holes.

STEP 8

Construct a sign chart to determine the behavior of the function across different intervals:
- Identify critical points: $x = -7$ , $x = -\frac{1}{2}$ , $x = 2$ , $x = 7$ . - Test intervals: $(- \infty, -7)$ , $(-7, -\frac{1}{2})$ , $(- \frac{1}{2}, 2)$ , $(2, 7)$ , $(7, \infty)$ .
Determine the sign of $f(x)$ in each interval by selecting test points and evaluating the sign of each factor.

STEP 9

Fill out the sign chart with the determined signs for each interval.

STEP 10

Sketch the function using the information gathered:
- Plot the horizontal asymptote $y = 4$ . - Plot the vertical asymptotes at $x = -7$ and $x = 2$ . - Mark the roots at $x = -\frac{1}{2}$ and $x = 7$ . - Use the sign chart to determine the behavior of the function in each interval and sketch the curve accordingly.
The function has horizontal asymptote $y = 4$ , vertical asymptotes at $x = -7$ and $x = 2$ , roots at $x = -\frac{1}{2}$ and $x = 7$ , and no holes. The sign chart helps to sketch the function accurately.

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