(a) Determine the equation of the line passing through the turning poin
of and the intersection point of the asymptotes

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Nov 07,2024

Question

(a) Determine the equation of the line passing through the turning poin of $$ y=-x^{2}+2 x+3 $$ and the intersection point of the asymptotes $$ y=-\frac{1}{x+2}-1 $$

UpStudy AI · Accepted Answer

To determine the equation of the line passing through the turning point of the quadratic function $$ y = -x^2 + 2x + 3 $$ and the intersection point of the asymptotes of the rational function $$ y = -\frac{1}{x+2} - 1 $$, we will follow these steps:

### Step 1: Find the turning point of the quadratic function

The turning point (vertex) of a quadratic function in the form $$ y = ax^2 + bx + c $$ can be found using the formula:

$$
x = -\frac{b}{2a}
$$

For the function $$ y = -x^2 + 2x + 3 $$:
- $$ a = -1 $$
- $$ b = 2 $$

Calculating the x-coordinate of the vertex:

$$
x = -\frac{2}{2 \cdot -1} = 1
$$

Now, substitute $$ x = 1 $$ back into the function to find the y-coordinate:

$$
y = -1^2 + 2(1) + 3 = -1 + 2 + 3 = 4
$$

Thus, the turning point is $$ (1, 4) $$.

### Step 2: Find the intersection point of the asymptotes

The function $$ y = -\frac{1}{x+2} - 1 $$ has a vertical asymptote at $$ x = -2 $$ (where the denominator is zero) and a horizontal asymptote at $$ y = -1 $$ (as $$ x \to \infty $$ or $$ x \to -\infty $$).

The intersection point of the asymptotes is simply the point where the vertical asymptote $$ x = -2 $$ intersects the horizontal asymptote $$ y = -1 $$:

$$
(-2, -1)
$$

### Step 3: Find the equation of the line through the two points

Now we have two points:
1. Turning point: $$ (1, 4) $$
2. Intersection point of the asymptotes: $$ (-2, -1) $$

To find the equation of the line passing through these two points, we first calculate the slope $$ m $$:

$$
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 4}{-2 - 1} = \frac{-5}{-3} = \frac{5}{3}
$$

Now, we can use the point-slope form of the equation of a line, which is given by:

$$
y - y_1 = m(x - x_1)
$$

Using the point $$ (1, 4) $$:

$$
y - 4 = \frac{5}{3}(x - 1)
$$

Now, simplify this equation:

$$
y - 4 = \frac{5}{3}x - \frac{5}{3}
$$

Adding 4 to both sides:

$$
y = \frac{5}{3}x - \frac{5}{3} + 4
$$

Convert 4 to a fraction with a denominator of 3:

$$
4 = \frac{12}{3}
$$

So,

$$
y = \frac{5}{3}x - \frac{5}{3} + \frac{12}{3} = \frac{5}{3}x + \frac{7}{3}
$$

### Final Answer

The equation of the line passing through the turning point of the quadratic function and the intersection point of the asymptotes is:

$$
y = \frac{5}{3}x + \frac{7}{3}
$$
The equation of the line is $$ y = \frac{5}{3}x + \frac{7}{3} $$.

(a) Determine the equation of the line passing through the turning poin
of and the intersection point of the asymptotes

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(a) Determine the equation of the line passing through the turning poin of and the intersection point of the asymptotes