Math / Algebra

QuestionQuestion 1 1 pts
Describe the vertical asymptotes and holes for the graph of $y=\frac{x+7}{x^{2}+5 x-14}$ If there are no asymtopotes or holes in the graph type none.
Verticlal Asymptotes: x or y $\square$ $=$ $\square$
Horizontal Asymptote: x or y $\square$ $=$ $\square$
Hole/s in the graph $x$ or $y$ $\square$ $\square$

Studdy Solution

STEP 1

1. The function $y = \frac{x+7}{x^2 + 5x - 14}$ is a rational function.
2. Vertical asymptotes occur where the denominator is zero and the numerator is non-zero.
3. Holes occur where both the numerator and the denominator are zero at the same point.

STEP 2

1. Factor the denominator.
2. Determine the vertical asymptotes.
3. Determine any holes in the graph.
4. Determine the horizontal asymptote.

STEP 3

Factor the denominator $x^2 + 5x - 14$ .
The expression can be factored as:
$x^2 + 5x - 14 = (x + 7)(x - 2)$

STEP 4

Determine the vertical asymptotes by setting the factored denominator equal to zero and solving for $x$ :
$(x + 7)(x - 2) = 0$
This gives the solutions:
$x + 7 = 0 \quad \Rightarrow \quad x = -7$ $x - 2 = 0 \quad \Rightarrow \quad x = 2$
These are the potential vertical asymptotes.

STEP 5

Check if the numerator $x + 7$ is zero at $x = -7$ or $x = 2$ .
For $x = -7$ :
$x + 7 = -7 + 7 = 0$
Since both the numerator and denominator are zero at $x = -7$ , there is a hole at $x = -7$ .
For $x = 2$ :
$x + 7 = 2 + 7 = 9$
The numerator is not zero at $x = 2$ , so there is a vertical asymptote at $x = 2$ .

STEP 6

Determine any holes in the graph.
From the previous step, we found a hole at $x = -7$ .

STEP 7

Determine the horizontal asymptote by comparing the degrees of the numerator and denominator.
The degree of the numerator $x + 7$ is 1, and the degree of the denominator $x^2 + 5x - 14$ is 2.
Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is:
$y = 0$
Vertical Asymptote: $x = 2$
Horizontal Asymptote: $y = 0$
Hole in the graph: $x = -7$

Was this helpful?

Studdy Solution

STEP 1

1. The function $y = \frac{x+7}{x^2 + 5x - 14}$ is a rational function.
2. Vertical asymptotes occur where the denominator is zero and the numerator is non-zero.
3. Holes occur where both the numerator and the denominator are zero at the same point.

STEP 2

1. Factor the denominator.
2. Determine the vertical asymptotes.
3. Determine any holes in the graph.
4. Determine the horizontal asymptote.

STEP 3

Factor the denominator $x^2 + 5x - 14$ .
The expression can be factored as:
$x^2 + 5x - 14 = (x + 7)(x - 2)$

STEP 4

Determine the vertical asymptotes by setting the factored denominator equal to zero and solving for $x$ :
$(x + 7)(x - 2) = 0$
This gives the solutions:
$x + 7 = 0 \quad \Rightarrow \quad x = -7$ $x - 2 = 0 \quad \Rightarrow \quad x = 2$
These are the potential vertical asymptotes.

STEP 5

STEP 6

Determine any holes in the graph.
From the previous step, we found a hole at $x = -7$ .

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. The function y=x+7x2+5x−14 y = \frac{x+7}{x^2 + 5x - 14} y=x2+5x−14x+7​ is a rational function.2. Vertical asymptotes occur where the denominator is zero and the numerator is non-zero.3. Holes occur where both the numerator and the denominator are zero at the same point.

STEP 2

1. Factor the denominator.2. Determine the vertical asymptotes.3. Determine any holes in the graph.4. Determine the horizontal asymptote.

STEP 3

Factor the denominator x2+5x−14 x^2 + 5x - 14 x2+5x−14.The expression can be factored as:x2+5x−14=(x+7)(x−2) x^2 + 5x - 14 = (x + 7)(x - 2) x2+5x−14=(x+7)(x−2)

STEP 4

STEP 5

STEP 6

Determine any holes in the graph.From the previous step, we found a hole at x=−7 x = -7 x=−7.

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The function $y = \frac{x+7}{x^2 + 5x - 14}$ is a rational function.
2. Vertical asymptotes occur where the denominator is zero and the numerator is non-zero.
3. Holes occur where both the numerator and the denominator are zero at the same point.

1. Factor the denominator.
2. Determine the vertical asymptotes.
3. Determine any holes in the graph.
4. Determine the horizontal asymptote.

Factor the denominator $x^2 + 5x - 14$ .
The expression can be factored as:
$x^2 + 5x - 14 = (x + 7)(x - 2)$

Determine any holes in the graph.
From the previous step, we found a hole at $x = -7$ .