Determine the vertex and direction of opening of the
parabola for the following quadratic equation:

a.) Vertex:
Opens: Upward
b.) Vertex:
Opens: Upward
c.) Vertex:
Opens: Downward
d.) Vertex:
Opens: Downward
(

Ask by Boone Chavez. in the United States

Jan 26,2025

Question

Determine the vertex and direction of opening of the
parabola for the following quadratic equation:

a.) Vertex:
Opens: Upward
b.) Vertex:
Opens: Upward
c.) Vertex:
Opens: Downward
d.) Vertex:
Opens: Downward
(

Ask by Boone Chavez. in the United States

Jan 26,2025

UpStudy AI · Accepted Answer

To determine the **vertex** and the **direction of opening** of the parabola given by the quadratic equation:

$$
y = x^2 - 10x + 7
$$

### Step 1: Find the Vertex

The vertex of a parabola in the form $$ y = ax^2 + bx + c $$ can be found using the formula:

$$
\text{Vertex } \left( h, k \right) \quad \text{where} \quad h = -\frac{b}{2a}, \quad k = y(h)
$$

For the given equation:
- $$ a = 1 $$
- $$ b = -10 $$
- $$ c = 7 $$

Calculate $$ h $$:

$$
h = -\frac{-10}{2 \times 1} = \frac{10}{2} = 5
$$

Now, calculate $$ k $$ by substituting $$ x = 5 $$ into the equation:

$$
k = (5)^2 - 10(5) + 7 = 25 - 50 + 7 = -18
$$

So, the **vertex** is $$ (5, -18) $$.

### Step 2: Determine the Direction of Opening

The direction in which the parabola opens is determined by the coefficient $$ a $$:

- If $$ a > 0 $$, the parabola **opens upward**.
- If $$ a < 0 $$, the parabola **opens downward**.

In this case, $$ a = 1 $$ which is positive, so the parabola **opens upward**.

### Conclusion

The correct answer is:

**b.) Vertex: $$ (5, -18) $$  
Opens: Upward**

**Answer:**  
**b.) Vertex: $$ (5,-18) $$**  
**Opens: Upward**
The vertex is at $$ (5, -18) $$ and the parabola opens upward.