Math / Algebra

QuestionRewrite $f(x)$ in factored form, but do not simplify: $f(x)=\frac{x+4}{4 x^{2}-x-3}=\square$ x-Intercepts Write intercepts as $(x, y)$ points: $\square$ y-Intercepts Write intercepts as $(x, y)$ points: $\square$ Vertical Asymptotes $\square$ $x=$
Horizontal Asymptotes $\square$ Oblique Asymptotes $\square$ Question Help: Video

Studdy Solution

STEP 1

1. The function $f(x) = \frac{x+4}{4x^2 - x - 3}$ is a rational function.
2. To rewrite in factored form, factor the numerator and the denominator.
3. Intercepts and asymptotes are determined by setting parts of the function equal to zero or analyzing limits.

STEP 2

1. Factor the numerator and the denominator.
2. Find x-intercepts by setting the numerator equal to zero.
3. Find y-intercepts by evaluating the function at $x = 0$ .
4. Determine vertical asymptotes by setting the denominator equal to zero.
5. Determine horizontal asymptotes by analyzing the degrees of the numerator and the denominator.
6. Determine oblique asymptotes if applicable.

STEP 3

Factor the numerator and the denominator.
Numerator: $x + 4$ is already factored.
Denominator: $4x^2 - x - 3$ .
To factor the quadratic $4x^2 - x - 3$ , we look for two numbers that multiply to $4 \times (-3) = -12$ and add to $-1$ . These numbers are $-4$ and $3$ .
Rewrite the middle term using these numbers: $4x^2 - 4x + 3x - 3$
Factor by grouping: $(4x^2 - 4x) + (3x - 3)$ $4x(x - 1) + 3(x - 1)$
Factor out the common factor: $(4x + 3)(x - 1)$
Thus, the factored form of $f(x)$ is: $f(x) = \frac{x+4}{(4x+3)(x-1)}$

STEP 4

Find x-intercepts by setting the numerator equal to zero:
$x + 4 = 0$ $x = -4$
The x-intercept is $(-4, 0)$ .

STEP 5

Find y-intercepts by evaluating the function at $x = 0$ :
$f(0) = \frac{0 + 4}{4(0)^2 - 0 - 3} = \frac{4}{-3} = -\frac{4}{3}$
The y-intercept is $(0, -\frac{4}{3})$ .

STEP 6

Determine vertical asymptotes by setting the denominator equal to zero:
$(4x + 3)(x - 1) = 0$
Solve for $x$ : $4x + 3 = 0 \quad \Rightarrow \quad x = -\frac{3}{4}$ $x - 1 = 0 \quad \Rightarrow \quad x = 1$
The vertical asymptotes are $x = -\frac{3}{4}$ and $x = 1$ .

STEP 7

Determine horizontal asymptotes by comparing the degrees of the numerator and the denominator.
The degree of the numerator is 1, and the degree of the denominator is 2.
Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$ .

STEP 8

Determine oblique asymptotes.
Since the degree of the numerator is less than the degree of the denominator, there are no oblique asymptotes.
The factored form of $f(x)$ is: $f(x) = \frac{x+4}{(4x+3)(x-1)}$
x-Intercepts: $(-4, 0)$
y-Intercepts: $(0, -\frac{4}{3})$
Vertical Asymptotes: $x = -\frac{3}{4}, x = 1$
Horizontal Asymptotes: $y = 0$
Oblique Asymptotes: None

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Studdy Solution

STEP 1

1. The function $f(x) = \frac{x+4}{4x^2 - x - 3}$ is a rational function.
2. To rewrite in factored form, factor the numerator and the denominator.
3. Intercepts and asymptotes are determined by setting parts of the function equal to zero or analyzing limits.

STEP 2

STEP 3

STEP 4

Find x-intercepts by setting the numerator equal to zero:
$x + 4 = 0$ $x = -4$
The x-intercept is $(-4, 0)$ .

STEP 5

Find y-intercepts by evaluating the function at $x = 0$ :
$f(0) = \frac{0 + 4}{4(0)^2 - 0 - 3} = \frac{4}{-3} = -\frac{4}{3}$
The y-intercept is $(0, -\frac{4}{3})$ .

STEP 6

Determine vertical asymptotes by setting the denominator equal to zero:
$(4x + 3)(x - 1) = 0$
Solve for $x$ : $4x + 3 = 0 \quad \Rightarrow \quad x = -\frac{3}{4}$ $x - 1 = 0 \quad \Rightarrow \quad x = 1$
The vertical asymptotes are $x = -\frac{3}{4}$ and $x = 1$ .

STEP 7

Determine horizontal asymptotes by comparing the degrees of the numerator and the denominator.
The degree of the numerator is 1, and the degree of the denominator is 2.
Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$ .

STEP 8

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Studdy Solution

STEP 1

1. The function f(x)=x+44x2−x−3 f(x) = \frac{x+4}{4x^2 - x - 3} f(x)=4x2−x−3x+4​ is a rational function.2. To rewrite in factored form, factor the numerator and the denominator.3. Intercepts and asymptotes are determined by setting parts of the function equal to zero or analyzing limits.

STEP 2

STEP 3

STEP 4

Find x-intercepts by setting the numerator equal to zero:x+4=0 x + 4 = 0 x+4=0 x=−4 x = -4 x=−4The x-intercept is (−4,0)(-4, 0)(−4,0).

STEP 5

Find y-intercepts by evaluating the function at x=0 x = 0 x=0:f(0)=0+44(0)2−0−3=4−3=−43 f(0) = \frac{0 + 4}{4(0)^2 - 0 - 3} = \frac{4}{-3} = -\frac{4}{3} f(0)=4(0)2−0−30+4​=−34​=−34​The y-intercept is (0,−43)(0, -\frac{4}{3})(0,−34​).

STEP 6

STEP 7

Determine horizontal asymptotes by comparing the degrees of the numerator and the denominator.The degree of the numerator is 1, and the degree of the denominator is 2.Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0 y = 0 y=0.

STEP 8

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1. The function $f(x) = \frac{x+4}{4x^2 - x - 3}$ is a rational function.
2. To rewrite in factored form, factor the numerator and the denominator.
3. Intercepts and asymptotes are determined by setting parts of the function equal to zero or analyzing limits.

Find x-intercepts by setting the numerator equal to zero:
$x + 4 = 0$ $x = -4$
The x-intercept is $(-4, 0)$ .

Find y-intercepts by evaluating the function at $x = 0$ :
$f(0) = \frac{0 + 4}{4(0)^2 - 0 - 3} = \frac{4}{-3} = -\frac{4}{3}$
The y-intercept is $(0, -\frac{4}{3})$ .

Determine horizontal asymptotes by comparing the degrees of the numerator and the denominator.
The degree of the numerator is 1, and the degree of the denominator is 2.
Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$ .